Influence of roughness height on the distribution of modeled turbulence statistics in non-aerated skimming flows in steep stepped spillways

Influence of roughness height on the distribution of modeled turbulence statistics in non-aerated skimming flows in steep stepped spillways | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Influence of roughness height on the distribution of modeled turbulence statistics in non-aerated skimming flows in steep stepped spillways Juan Pablo Toro, Alex Blanc, Patricio Moreno-Casas, Sebastián Sepúlveda This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3538512/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The stepped spillway design is widely used globally for its energy dissipation capabilities through its stepped design and air entrainment capacity, crucial for safe hydraulic structure operation. The most common condition found in stepped spillways is the skimming flow regime. This study numerically investigated the influence of the roughness height (K S ) on the distribution of mean flow variables and modeled turbulence statistics in the non-aerated portion of a steep stepped spillway with a constant angle of 51.34°. Three roughness heights of 6.26, 3.13 and 1.57 cm were considered corresponding to a relation step height/horizontal length of 10/8, 5/4 and 2.5/2 (cm/cm), respectively. The results show that profiles of modeled turbulent kinetic energy (TKE) at different step edges can be collapsed into one single curve for relative roughness (Ks/h) smaller than 0.8. For larger ratios of roughness height to water depth, this behavior is not observed and for a given distance away from the wall, values of TKE show a broad range of values, with increasing values as boundary layer grows. A similar trend was obtained for the distribution of TKE along a water column that includes the cavity. The existence of an upper limit for the ratio between the roughness height and the water depth can now be incorporated in the hydraulic design of new steep stepped spillways. Stepped spillway Self-similarity Roughness height Boundary layer Turbulent kinetic energy Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1 Introduction A time-dependent flow solution is called self-similar if spatial distributions of its properties at different times can be obtained from one another by a similarity transformation (Barenblatt and Isaakovich 1996 ). From a fluid mechanics perspective, self-similarity of mean flow variables and turbulence statistics is an appealing feature of turbulent flows, since profiles of such distributions can be typically collapsed into one single curve (Hassan et al. 2011 ). These universal solutions are also attractive from the design and civil engineering point of view. Self-similarity appears in boundary-layer flows including jets (Kuhn et al. 2021 ) and plumes (Cenedese and Linden 2014 ); flows past hydraulic structures (Lin et al. 2012 ) and airfoils (DeVoria and Mohseni 2021 ); pipe (Hellström et al. 2016 ) and open channel flows (Ercan et al. 2014 ), and turbulence in rotating fluids (Baroud et al. 2002 ), among others (Heller 2017 ; Kartal and Emiroglu, 2023 ; Giudicianni et al. 2021 ). In a very recent numerical study, self-similarity of turbulent stresses has been reported in a compressible flow over a convex wall (Wang et al. 2023 ). Guinot et al. ( 2023 ) have also documented self-similar distributions in experimentally obtained block-averaged water depths during urban flood events. In the realm of flood control, it is imperative to conduct experimental assessments of flow characteristics within physical models of hydraulic structures, such as spillways and stepped chutes (see for example Eghlidi et al. 2020 ; Foroudi et al. 2022 ). These measurements play a pivotal role as they enable engineers to enhance their design criteria and validate numerical simulations using the resultant dataset of flow characteristics. The principle of self-similarity has also been reported in this type of hydraulic structures. For example, in the aerated portion of stepped chutes with mild slopes, profiles of time-averaged interfacial velocity and air concentration have been found to have the same shape, independent of space (Chanson and Carosi 2007 ; Gonzalez et al. 2008 ). Self-similarity, however, has not yet been scrutinized in boundary-layer channel flows with steep slopes and large surface-roughness elements such as those present in steep stepped spillways. The current numerical study aims to answer this question by analyzing the influence of the roughness height of a steep stepped spillway on the mean flow variables and turbulence statistics in the non-aerated region of a skimming flow. The skimming flow (Rajaratnam 1990 ) is the characteristic flow regime over steep stepped spillways. In the upstream portion of a skimming flow, the water surface is smooth and glassy (Wood 1991 ) and no significant amount of air is entrained through the water surface. Several empirical relationships have been proposed to estimate the distance from the crest of the stepped spillway to the inception point of air entrainment (i.e., the length of the non-aerated flow region). They all depend on the Froude number based on the roughness height as characteristic length scale \({F}_{*}=q/{\left(gsin\left(\theta \right){{k}_{s}}^{3}\right)}^{1/2}\) ( \(q=\) flowrate per unit width, \(g=\) acceleration of gravity, \(\theta =\) angle between the pseudobottom formed by the step edges and the horizontal, \({k}_{s}={h}_{s} cos\left(\theta \right)=\) step dimension measured normal to the flow and \({h}_{s}=\) height of steps). The current numerical study gives new evidence of the upper limit of the ratio among the roughness height and the water depth, for which distributions of modeled turbulent kinetic energy in the non-aerated region of steep stepped spillways follow a self-similar distribution. 2 Computational Setup Stepped spillways of variable step dimensions were utilized in the numerical experiments and the geometrical details are presented in Table 1 . Except for the size of the steps, the three modeled stepped spillways have identical geometry (Fig. 1 ). The step dimensions for cases A and B were chosen so the angle between the pseudo-bottom formed by the step edges and the horizontal was constant and equal to that of Base case, \(\theta ={51.34}^{^\circ }\) . The roughness height corresponds to the maximum step dimension measured normal to the flow \(\left({k}_{s}\right)\) , whereas \({{k}_{s}}_{50\%}\) is defined herein as the roughness height measured normal to the flow from the middle of the diagonal of steps (Fig. 1 ). Note that in each of the three stepped spillways, the same four cross sections identified by the names CS1, CS2, CS3, and CS4 are defined, which correspond precisely to step edges. Additionally, for each spillway there are also four extra sections located in the middle of the cavity, immediately downstream of each step edge that belongs to each selected cross section. A constant flow rate per unit width of \(q=0.11\) m 2 /s was employed for all spillways. The Reynolds number for all cases was \(Re=q/\nu \sim{10}^{5}\) allowing the use of large-Reynolds-based turbulence closures. Since the runs are two-dimensional (2D), the number of computational cells is small in all three cases and close to 90,000 cells. In this work, it was assumed that the position of the inception point is located further downstream outside the computational domain implemented in all three stepped spillways. This assumption is valid, since for Base case, we have experimental measurements that validate this (Amador 2005 ); for case B we checked that all empirical relationships presented in Chanson ( 1994 ), Matos ( 1999 ), Sánchez-Juny ( 2001 ), Boes and Hager ( 2003 ), Amador et al. ( 2009 ), and Meireles et al. ( 2012 ) predict a larger non-aerated region than that of Base case. For case A, empirical relationships predict a non-aerated region, which on average is ~ 9.2 cm smaller than that obtained for Base case, which still allows us to be in the non-aerated flow region. This is because the hypotenuse for one step in case A is 12.8 cm only. Further, we are not presenting results for the last step edge in case A. Numerical works associated to the geometry of Base case have been successfully validated with experimental data (Toro et al. 2016 ; Toro et al. 2017 ), whereas cases A and B are the selected variants of Base case to assess the influence of the roughness height. Table 1 Geometric details of the three modeled stepped spillways Case Height of steps \({h}_{s}\) (cm) Horizontal length of steps (cm) Step dimension measured normal to the flow, \({k}_{s}\) (cm) \({{k}_{s}}_{50\%}\) (cm) Number of cells A 10 8 6.25 5.12 93,042 Base 5 4 3.12 2.56 89,794 B 2.5 2 1.56 1.28 88,364 The computational domain is discretized using cells of size \({\Delta }x={\Delta }y=2 mm\) , and the associated average \({y}^{+}\) values over the steps are 48, 49, and 38, for cases A, Base and B, respectively. Boundary conditions are the same as those presented in Toro et al. ( 2016 ). The employed divergence scheme for both turbulent kinetic energy (TKE), \(k\) , and dissipation rate of TKE, \(\epsilon\) , was the linear scheme Gauss limitedLinear . For the effective viscosity, the second order Gauss linear scheme was utilized. Gauss vannLeer and Gauss interfaceCompression were employed for the divergence terms appearing in the transport equation of the volume fraction occupied by the water. For Laplacian schemes, the Gauss linear corrected method was applied. 3 Results and Discussion The results below correspond to those obtained for a fully developed flow, which occurs after 4 s of simulation. It was also verified that there were no regions with air close to the cavities, since only the non-aerated region of the spillway was simulated. In Fig. 1 , the solution of the developed flow is presented for each spillway. 3.1 Mean Flow Velocity In Fig. 2 , the distribution of mean flow velocities at four step edges for the three stepped spillways studied, is presented. It can be clearly observed that the flow velocity increases as the flow develops along the spillway. At the closest step edge to the crest (CS1), the flow velocities are in the vicinity of 2.6 m/s for all cases, while the distribution is nearly the same, as expected, since at this point the flow is still unaware of the roughness imposed by the steps located further downstream, and the geometry in the upstream portion is the same for the three spillways, except for the short length of 12.8 cm, located immediately upstream of CS1. One can note that in all cases the maximum velocity at CS4 is close to 3.5 m/s, although minor differences among the three spillways are explained by the growth of the velocity within the boundary layer. In CS4 is observed that, at any height within the boundary layer, the stepped spillway with the smallest roughness (case B) achieves the highest velocities and the smallest boundary layer thickness, thus indicating that the flow in the water column forgets the interaction with the cavity faster when the roughness height is smaller. It is also noticeable that to comply with the principle of mass conservation, slightly higher water depths are observed in case A. 3.2 Turbulent Kinetic Energy In Fig. 3 , the normalized values of modeled turbulent kinetic energy (TKE), within the boundary layer, at four step edges for all spillways are presented. TKE values are made non-dimensional by using the corresponding maximum free-stream velocity at the step edges. $${k}_{norm}=k/{{V}_{max}}^{2}$$ where \({k}_{norm}\) represents the normalized modeled TKE, \(k\) depicts the TKE, and \({V}_{max}\) is the maximum flow velocity in the water column. According to Fig. 3 , the profiles of TKE for each spillway follow a clear self-similar pattern, except for case A, which is the case with the largest roughness height. This can be attributed to the roughness layer covering the entire flow depth and not a portion of it. This is particularly relevant since it provides evidence of the existence of a maximum roughness height for which the distribution of TKE can be expressed through a unique curve. In other words, this would indicate the existence of an upper limit for the size of the cavity ( \({k}_{s}\) ) for which distributions of TKE at step edges can be collapsed into one single curve. It can also be observed that values of TKE are larger in the stepped spillway with larger roughness height, reaching maximum values ranging between 0.033 and 0.037, which are located at a height between 10 and 15% of the thickness of the boundary layer. For cases Base and B, maximum values of TKE are approximately 0.028 and 0.023 respectively, and they are both located at an approximate height of 15% of the thickness of the boundary layer. In Fig. 4 , the TKE values in the water column at the middle of the steps, corresponding to a roughness height of \({{k}_{s}}_{50\%}\) , are presented. Remarkably, just as the distributions of TKE at step edges (previous Fig. 3 ), cases B and Base present TKE values that can be reasonably well grouped by means of a single curve. This is different from case A, which presents greater variability in TKE values, especially at relative water depths between 0.5 and 0.8; this is, from just below the pseudobottom to above the pseudobottom. Inside the cavity, the values are well grouped for case A. Maximum values of TKE are located above the pseudobottom with representative values of 0.019, 0.022 and 0.027 for cases B, Base and A, respectively. 3.3 Boundary layer thickness \(\varvec{\delta }\) and relative roughness ( \({\varvec{k}}_{\varvec{s}}/\varvec{h}\) ) The growth of the boundary layer thickness ( \(\delta\) ) and the distribution of relative roughness ( \({k}_{s}/h\) ) for the three spillways and four cross sections are presented in Fig. 5 . Noteworthy is the similar rate of growth of the boundary layer for the three stepped spillways between cross sections 1 to 4 (Fig. 5 a). This corresponds to the approximate slope of the line connecting the thickness of the boundary layer in those four points. However, in cross section 1 , for both cases, Base and B, the boundary layer thickness is approximately 1.6 cm, whereas in case A the value is larger and equal to 2.4 cm. This is because in case A the flow recognizes more quickly the existence of a larger roughness height. It can be inferred from Fig. 5 b that only in cases B and Base, water depths obtained at step edges are a fraction of the size of the roughness element ( \({k}_{s}/h\) smaller than ~ 0.8). This value will be considered as the upper limit to obtain self-similar distributions of flow variables. 3.4 Rate of dissipation of turbulent kinetic energy The normalized rate of dissipation of turbulent kinetic energy ( \({\epsilon }_{norm}\) ) can be described as: $${\epsilon }_{norm}=\frac{\epsilon \delta }{{V}_{max}^{3}} \left(1\right)$$ where \(\epsilon\) is the rate of dissipation of TKE and \(\delta\) is the boundary layer thickness. The \({\epsilon }_{norm}\) values computed from the three simulated steeped spillways at the selected four cross sections are shown in Fig. 6 . It is very clear that the peak rate of dissipation of energy occurs in proximity to the step edges at each selected cross section, progressively decreasing as distance from the pseudobottom increases. For all cases (Base, A and B) and cross sections (CS1-CS4) the curves of \(\epsilon\) describe a pseudo self-similar shape. This is because a self-similar distribution of \(\epsilon\) values is observed only for \(y/\delta >0.2\) . Indeed, it is evident that close to the step edges ( \(y/\delta <0.2\) ), there is a range of maximum values of epsilon and therefore a unique curve cannot be employed to summarize those values. Perhaps it is worth emphasizing here that the rate of dissipation of turbulent kinetic energy is a highly challenging variable to model, as evident from its very definition. Strictly speaking, only a characterization of resolved turbulent flow fluctuations would enable the estimation of this variable. Therefore, what is described here pertains more to the mathematical behavior of the flow solution near the pseudobottom. For the spillway with smaller steps, case B, the self-similarity is more evident for \(y/\delta >0.05\) . In that case, epsilon values are well grouped, and therefore the rate of dissipation of TKE can be described by a single curve, independent of the step (cross section). Since case B presents the smaller roughness, its behavior better resembles the flow of conventional open channel flows. The maximum values of modeled rates of dissipation for case B are found right at the pseudobottom, with values ranging from 0.015 (CS1) to 0.017 (CS4), which indicates that the rate of dissipation is nearly constant at the pseudobottom along the spillway. Slightly larger differences can be found at the pseudobottom for Base case, where rates of dissipation range from 0.018 (CS1) to 0.026 (CS4). However, up, and not too far from the steps, \(y/\delta >0.10\) , the behavior of the rate of dissipation becomes independent of the cross section, and therefore self-similar. Finally, for the larger roughness height, case A, self-similarity of the rate of dissipation (independent of the location of the step) occurs at \(y/\delta \sim0.2\) and above, since below this height, the values of the rate of dissipation become dependent of the flow development as the fluid moves down the stepped spillway. The rate of dissipation values at the pseudobottom range from 0.025 (CS1) to 0.034 (CS4) for case A. From the above discussion it is possible to recognize that the height at which the rate of dissipation of TKE starts to be self-similar is directly correlated to the roughness height, since the larger the size of the cavity the larger the height at which self-similar behavior starts to occur. In a prior study conducted by Toro et al. ( 2016 ), the presence of self-similarity in turbulent kinetic energy (TKE) and the rate of dissipation of TKE for the Base case was demonstrated. However, it is important to remark that this investigation did not consider variations in roughness heights. 3.5 Mean pressure Dimensionless pressures ( \({p}^{*}\) ) at step edges and within the cavities are plotted in Figs. 7 and 8 , respectively. Pressures ( \(p\) ) were made dimensionless by using the projection of the water depth on the vertical and they were calculated as: $${p}^{*}=\frac{p}{\gamma hcos\left(\theta \right)} \left(1\right)$$ Where \(\gamma\) is the specific weight of water (N/m 3 ), \(h\) is the flow depth measured normal to the spillway pseudobottom, and \(\theta\) is the slope of the stepped spillway. Since the denominator in the above equation, \(\gamma hcos\left(\theta \right)\) , represents the hydrostatic pressure, \({p}^{*}\) values above, equal, and below one, indicate pressures over, equal to or less than the hydrostatic pressure, respectively. In Fig. 7 it is shown that from the water surface down to about 40% of the dimensionless water depth, case B has a pressure distribution at step edges which is remarkably like the hydrostatic pressure distribution, attributed to the small cavity. Hence, the pressure field is affected by the influence of the cavity only in the first 40% of the water column, reaching maximum values varying from 1.5 to 2. For cases A and Base, pressure distributions at step edges remain very distant from the hydrostatic distribution. This is especially true for case A, due to the larger roughness height. In case A, the highest-pressure values are located approximately at 20% of the water depth, with a range of values between 1.4 (CS1) and 2.2 (CS2). Finally, for Base case, maximum pressures are located close to step edges as in case B, with values varying from 1.5 to 2.5. In Fig. 8 , the dimensionless pressure inside the step cavities and extending up to a distance of \(\delta\) above the pseudobottom is depicted for various cross sections. Pressure values are thus analyzed from the cavity wall ( \(Y=0\) ) up to \(Y=K{s}_{50\%}+\delta\) (see Fig. 1 ) since the focus is on the relationship between the flow in the cavity and the flow in the shear layer, just above the pseudobottom. Case B, with the smallest roughness height, is the only case in which pressure values inside the cavity are larger than those obtained from the hydrostatic distribution, for all cross sections studied. Pressure profiles resemble an S-shaped curve where two inflection points can be identified: one at \({Y}^{*}\sim0.08\) and another at \({Y}^{*}\sim0.30\) . For \({Y}^{*}>0.40\) , and moving towards the free surface, all curves follow a quasi-linear distribution. On the contrary, cases A and Base, with larger roughness heights, exhibit pressures below the hydrostatic pressure, attributed to the recirculating flow inside the cavity. This provides evidence that the pressure distribution within the cavity ceases to be hydrostatic once a particular roughness height is exceeded. Below the roughness size threshold, and due to the lack or minimum recirculation within the cavity, the pressure profile is always larger than hydrostatic. As the cavity becomes larger, flow recirculation grows, decreasing pressure below the hydrostatic pressure. For the Base case two inflection points can be identified: the first one is located near \({Y}^{*}\sim0.30\) , where minimum pressure values ( \({P}^{*}\) ) range from − 0.8 (CS4) up to -0.35 (CS1). The second inflection point is around \({Y}^{*}\sim0.85\) . In case A with the largest cavity, on the other hand, only one inflection point is observed, close to \({Y}^{*}\sim0.40\) , where the pressure reaches its minimum value for all four cross sections. It is also observed that the initial two cross sections CS1 and CS2, exhibit a behavior that closely resembles a single curve with a minimum pressure value around − 0.25. This drastically changes for profiles in cross sections CS3 and CS4, where minimum pressures move to about − 0.55 and − 0.7, respectively. Cases A and Base share the feature of pressure distributions lower than the hydrostatic counterpart. They also share that further downstream, the minimum pressure values within the cavity are progressively decreasing. For the larger roughness height (case A), however, the pressure is distributed more smoothly than in Base case, which presents a more pronounced pressure peak at the lowest inflection point. 4 Conclusions The present work corresponds to a numerical study of the flow over steep stepped spillways, with the emphasis placed on the effect of the roughness height on the distribution of the mean flow velocity and pressure, and distribution of modeled turbulence statistics, namely the turbulent kinetic energy (TKE) and rate of viscous dissipation ( \(\epsilon\) ). To that end, three cases were analyzed. Case A, with a step height of 10 cm and a horizontal length of 8 cm ( \({k}_{s}\) =6.25 cm) which corresponds to the larger cavity dimensions; Base case with half the dimensions of case A ( \({k}_{s}\) =3.12 cm) and case B, with half the dimensions of Base case ( \({k}_{s}\) =1.56 cm). From the obtained results, the following conclusions can be extracted: For Cases B and Base, which are the stepped spillways with the smaller steps, distribution of turbulent kinetic energy within the boundary layer at the step edges can be summarized in one single curve. This self-similarity behavior is not observed for Case A, which has the larger steps. According to this, it might be argued that self-similarity for the modeled values of TKE is met only for values \({k}_{s}/h<0.8\) which correspond to the upper limit for cases B and Base analyzed in this study. The greater the size of the cavity, the greater the maximum velocities inside the cavities. Maximum velocities inside the cavities for cases A, Base and B are: 0,89; 0.76; y 0.28 m/s, respectively. For a given location, it was found that the larger the roughness height is, the larger the thickness of the boundary layer associated. For example, at cross section CS4, boundary layer thicknesses of 2.8, 2.4 and 2.1 cm were obtained for cases A, Base and B, respectively. However, the rate of growth of the boundary layer for case A with greater steps is like that of case B with the smaller steps. Regardless of the size of the step cavity, maximum velocities in the water column are nearly the same. The velocity distribution at step edges for the three stepped spillways can be collapsed in one single curve. The flow velocity at the first control step is 2.6 m/s and it grows downstream up to 3.6 m/s at the fourth control step. Maximum values of TKE at step edges are located at 12% of the boundary layer thickness with a representative non-dimensional value of 0.037. Pressure distributions at step edges follow a distribution greater than the hydrostatic for the three studied cases. For case A, with greater steps, maximum values of pressure are located at 20% of the boundary layer, whereas in cases B and Base, maximum values for pressure are located close to step edges. Statements and Declarations Funding The first author gratefully acknowledges financial support of the National Fund for Scientific and Technological Development (FONDECYT) through ANID Iniciación en Investigación Project N° 11221196. Competing Interests The authors have no financial or non-financial interests to disclose. Author Contributions Juan Pablo Toro contributed to the study conception and design. Implementation of the numerical simulations and analysis were performed by Juan Pablo Toro and Alex Blanc. Figures preparation and data collection was performed by Alex Blanc and Sebastián Sepúlveda. The first draft of the manuscript was written by Juan Pablo Toro. Patricio Moreno-Casas contributed to the manuscript design and commented on previous versions of the manuscript. 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Journal of Hydraulic Engineering 116:587-591. https://doi.org/10.1061/(ASCE)0733-9429(1990)116:4(587) Sánchez-Juny M (2001) Comportamiento hidráulico de los aliviaderos escalonados en presas de hormigón compactado. Análisis del campo de presiones. Ph.D. thesis, Technical Univ. of Catalonia (UPC), Barcelona, Spain (in Spanish) Toro JP, Bombardelli FA, Paik J, Meireles I, Amador A (2016) Characterization of turbulence statistics on the non-aerated skimming flow over stepped spillways: a numerical study. Environmental Fluid Mechanics 16:1195–1221. https://doi.org/10.1007/s10652-016-9472-1 Toro JP, Bombardelli FA, Paik J (2017) Detached eddy simulation of the nonaerated skimming flow over a stepped spillway. Journal of Hydraulic Engineering 143:04017032. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001322 Wang Q, Qu F, Sun D, Bai J (2023) Numerical study of instabilities and compressibility effects on supersonic jet over a convex wall. Journal of Fluid Mechanics 954:A6. https://doi.org/10.1017/jfm.2022.977 Wood IR (1991) Free surface air entrainment on spillways. In: Air entrainment in free-surface flows. A.A. Balkema, Rotterdam, The Netherlands, pp 55-84 Supplementary Files Highlights.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3538512","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":244976017,"identity":"9ae2c15a-6c7b-4d67-8a35-f89403737e7c","order_by":0,"name":"Juan Pablo Toro","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuklEQVRIiWNgGAWjYBACxgYwZSEHIg88IE4LM4iSMAZrSSDOHoiWRLBtRGlhnt1/8HFFhUT6/LDDD4G22MnpNhBy2JzDzIZnzkjkbrydZgDUkmxsdoCQlhnJbJKNbUAtsxNAWg4kbiNWS7rh7PQPpGlJkJfOIdaWOYeNDRvOSBhukM4pOJBgQIRfDGc3PnzYUGEjLz87ffOHDxV2coS1zIAyDMAqDQgoBwF5CRijgQjVo2AUjIJRMDIBAAtQQ5s2CgI9AAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0003-4961-2086","institution":"Universidad Andres Bello","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Juan","middleName":"Pablo","lastName":"Toro","suffix":""},{"id":244976018,"identity":"c13010d3-b5c5-4b0d-991a-b8c0c8af781c","order_by":1,"name":"Alex Blanc","email":"","orcid":"","institution":"Universidad Andres Bello","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Alex","middleName":"","lastName":"Blanc","suffix":""},{"id":244976019,"identity":"4992389f-41b5-4a69-b685-85f45c28f490","order_by":2,"name":"Patricio Moreno-Casas","email":"","orcid":"","institution":"Universidad de los Andes Facultad de Ingeniería y Ciencias Aplicadas: Universidad de los Andes Facultad de Ingenieria y Ciencias Aplicadas","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Patricio","middleName":"","lastName":"Moreno-Casas","suffix":""},{"id":244976020,"identity":"0ad432f0-e8f1-4805-a571-1c32e9fe3c22","order_by":3,"name":"Sebastián Sepúlveda","email":"","orcid":"","institution":"Universidad de los Andes Facultad de Ingeniería y Ciencias Aplicadas: Universidad de los Andes Facultad de Ingenieria y Ciencias Aplicadas","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Sebastián","middleName":"","lastName":"Sepúlveda","suffix":""}],"badges":[],"createdAt":"2023-11-01 21:25:44","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3538512/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3538512/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":46009127,"identity":"d2e20781-1c54-4492-8eee-43778ab4e53f","added_by":"auto","created_at":"2023-11-07 13:46:08","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":176158,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic of the stepped spillways geometry and the selected four cross sections (CS) and details of the roughness height, Ks and Ks_(50%). From top to bottom; cases Base, A, and B, respectively\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/c31d534223fb1e9e3a70d4f4.png"},{"id":46009125,"identity":"1365f63a-7de7-41d9-b2d2-e3430a29c7cc","added_by":"auto","created_at":"2023-11-07 13:46:08","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":72611,"visible":true,"origin":"","legend":"\u003cp\u003eMean flow velocity profiles at four step edges\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/80b2afb86d14bb18d9eceda5.png"},{"id":46009123,"identity":"8929dffd-c462-4751-9ecc-e9521ce95de0","added_by":"auto","created_at":"2023-11-07 13:46:08","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":82314,"visible":true,"origin":"","legend":"\u003cp\u003eProfiles of turbulent kinetic energy (TKE) at four step edges\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/6fde9ad3330b8e182581b0b9.png"},{"id":46009656,"identity":"9fd781da-52ea-4832-968d-6b7bde6bf01d","added_by":"auto","created_at":"2023-11-07 13:54:08","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":161303,"visible":true,"origin":"","legend":"\u003cp\u003eProfiles of turbulent kinetic energy (TKE) in water columns that include cavities (ks50%)\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/6c4bf5b6cda3406190548222.png"},{"id":46009655,"identity":"744af12c-8652-44a3-8bf5-61e0338c107b","added_by":"auto","created_at":"2023-11-07 13:54:08","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":32874,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ea\u003c/strong\u003e Thickness of the boundary layer, \u003cstrong\u003eb\u003c/strong\u003e Relative roughness (roughness height/water depth)\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/2a5f4860a3dbe5f8bd1eb804.png"},{"id":46009129,"identity":"f23eb6eb-556f-4247-90b8-6eb045bf8bb9","added_by":"auto","created_at":"2023-11-07 13:46:08","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":101781,"visible":true,"origin":"","legend":"\u003cp\u003eProfiles of dimensionless rate of dissipation of TKE at selected step edges for cases A, B and Base\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/0aee70c609a82552d477a2c8.png"},{"id":46010372,"identity":"d398ccff-1e9b-4361-92ef-8d3867379c95","added_by":"auto","created_at":"2023-11-07 14:02:08","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":165391,"visible":true,"origin":"","legend":"\u003cp\u003eProfiles of mean pressure at step edges\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/7a6b11b307ad00d4dd2cbb25.png"},{"id":46009131,"identity":"27e84ddf-e133-4510-8de6-c78ffe53c56a","added_by":"auto","created_at":"2023-11-07 13:46:08","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":145657,"visible":true,"origin":"","legend":"\u003cp\u003eProfiles of mean pressure in water columns that include cavities (ks\u003csub\u003e50%\u003c/sub\u003e). Horizontal lines corresponding to pseudobottom of cases A, Base, and B are also shown.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/c58c236712837d264279abd8.png"},{"id":61731285,"identity":"9e64f6a9-7fee-41be-ab49-4dfb2efa7e23","added_by":"auto","created_at":"2024-08-05 00:58:31","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1219526,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/943a4249-3fde-4c86-84fc-c589be634c2c.pdf"},{"id":46009124,"identity":"f6e20855-6956-4961-8dc1-1e61ad649b19","added_by":"auto","created_at":"2023-11-07 13:46:08","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":15641,"visible":true,"origin":"","legend":"","description":"","filename":"Highlights.docx","url":"https://assets-eu.researchsquare.com/files/rs-3538512/v1/04036301006fd081b714d23d.docx"}],"financialInterests":"","formattedTitle":"Influence of roughness height on the distribution of modeled turbulence statistics in non-aerated skimming flows in steep stepped spillways","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eA time-dependent flow solution is called self-similar if spatial distributions of its properties at different times can be obtained from one another by a similarity transformation (Barenblatt and Isaakovich \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e1996\u003c/span\u003e). From a fluid mechanics perspective, self-similarity of mean flow variables and turbulence statistics is an appealing feature of turbulent flows, since profiles of such distributions can be typically collapsed into one single curve (Hassan et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). These universal solutions are also attractive from the design and civil engineering point of view. Self-similarity appears in boundary-layer flows including jets (Kuhn et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) and plumes (Cenedese and Linden \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2014\u003c/span\u003e); flows past hydraulic structures (Lin et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2012\u003c/span\u003e) and airfoils (DeVoria and Mohseni \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2021\u003c/span\u003e); pipe (Hellstr\u0026ouml;m et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) and open channel flows (Ercan et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2014\u003c/span\u003e), and turbulence in rotating fluids (Baroud et al. \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2002\u003c/span\u003e), among others (Heller \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Kartal and Emiroglu, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Giudicianni et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). In a very recent numerical study, self-similarity of turbulent stresses has been reported in a compressible flow over a convex wall (Wang et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Guinot et al. (\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) have also documented self-similar distributions in experimentally obtained block-averaged water depths during urban flood events.\u003c/p\u003e \u003cp\u003eIn the realm of flood control, it is imperative to conduct experimental assessments of flow characteristics within physical models of hydraulic structures, such as spillways and stepped chutes (see for example Eghlidi et al. \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Foroudi et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). These measurements play a pivotal role as they enable engineers to enhance their design criteria and validate numerical simulations using the resultant dataset of flow characteristics. The principle of self-similarity has also been reported in this type of hydraulic structures. For example, in the aerated portion of stepped chutes with mild slopes, profiles of time-averaged interfacial velocity and air concentration have been found to have the same shape, independent of space (Chanson and Carosi \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Gonzalez et al. \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). Self-similarity, however, has not yet been scrutinized in boundary-layer channel flows with steep slopes and large surface-roughness elements such as those present in steep stepped spillways. The current numerical study aims to answer this question by analyzing the influence of the roughness height of a steep stepped spillway on the mean flow variables and turbulence statistics in the non-aerated region of a skimming flow.\u003c/p\u003e \u003cp\u003eThe skimming flow (Rajaratnam \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e1990\u003c/span\u003e) is the characteristic flow regime over steep stepped spillways. In the upstream portion of a skimming flow, the water surface is smooth and glassy (Wood \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e1991\u003c/span\u003e) and no significant amount of air is entrained through the water surface. Several empirical relationships have been proposed to estimate the distance from the crest of the stepped spillway to the inception point of air entrainment (i.e., the length of the non-aerated flow region). They all depend on the Froude number based on the roughness height as characteristic length scale \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F}_{*}=q/{\\left(gsin\\left(\\theta \\right){{k}_{s}}^{3}\\right)}^{1/2}\\)\u003c/span\u003e\u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(q=\\)\u003c/span\u003e\u003c/span\u003eflowrate per unit width, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(g=\\)\u003c/span\u003e\u003c/span\u003eacceleration of gravity, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta =\\)\u003c/span\u003e\u003c/span\u003e angle between the pseudobottom formed by the step edges and the horizontal, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}={h}_{s} cos\\left(\\theta \\right)=\\)\u003c/span\u003e\u003c/span\u003e step dimension measured normal to the flow and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h}_{s}=\\)\u003c/span\u003e\u003c/span\u003e height of steps).\u003c/p\u003e \u003cp\u003eThe current numerical study gives new evidence of the upper limit of the ratio among the roughness height and the water depth, for which distributions of modeled turbulent kinetic energy in the non-aerated region of steep stepped spillways follow a self-similar distribution.\u003c/p\u003e"},{"header":"2 Computational Setup","content":"\u003cp\u003eStepped spillways of variable step dimensions were utilized in the numerical experiments and the geometrical details are presented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Except for the size of the steps, the three modeled stepped spillways have identical geometry (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The step dimensions for cases A and B were chosen so the angle between the pseudo-bottom formed by the step edges and the horizontal was constant and equal to that of Base case, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta ={51.34}^{^\\circ }\\)\u003c/span\u003e\u003c/span\u003e. The roughness height corresponds to the maximum step dimension measured normal to the flow \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({k}_{s}\\right)\\)\u003c/span\u003e\u003c/span\u003e, whereas \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{k}_{s}}_{50\\%}\\)\u003c/span\u003e\u003c/span\u003e is defined herein as the roughness height measured normal to the flow from the middle of the diagonal of steps (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Note that in each of the three stepped spillways, the same four cross sections identified by the names CS1, CS2, CS3, and CS4 are defined, which correspond precisely to step edges. Additionally, for each spillway there are also four extra sections located in the middle of the cavity, immediately downstream of each step edge that belongs to each selected cross section.\u003c/p\u003e \u003cp\u003eA constant flow rate per unit width of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(q=0.11\\)\u003c/span\u003e\u003c/span\u003e m\u003csup\u003e2\u003c/sup\u003e/s was employed for all spillways. The Reynolds number for all cases was \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Re=q/\\nu \\sim{10}^{5}\\)\u003c/span\u003e\u003c/span\u003e allowing the use of large-Reynolds-based turbulence closures. Since the runs are two-dimensional (2D), the number of computational cells is small in all three cases and close to 90,000 cells. In this work, it was assumed that the position of the inception point is located further downstream outside the computational domain implemented in all three stepped spillways. This assumption is valid, since for Base case, we have experimental measurements that validate this (Amador \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2005\u003c/span\u003e); for case B we checked that all empirical relationships presented in Chanson (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1994\u003c/span\u003e), Matos (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1999\u003c/span\u003e), S\u0026aacute;nchez-Juny (\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2001\u003c/span\u003e), Boes and Hager (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2003\u003c/span\u003e), Amador et al. (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2009\u003c/span\u003e), and Meireles et al. (\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2012\u003c/span\u003e) predict a larger non-aerated region than that of Base case. For case A, empirical relationships predict a non-aerated region, which on average is ~\u0026thinsp;9.2 cm smaller than that obtained for Base case, which still allows us to be in the non-aerated flow region. This is because the hypotenuse for one step in case A is 12.8 cm only. Further, we are not presenting results for the last step edge in case A. Numerical works associated to the geometry of Base case have been successfully validated with experimental data (Toro et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Toro et al. \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), whereas cases A and B are the selected variants of Base case to assess the influence of the roughness height.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eGeometric details of the three modeled stepped spillways\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCase\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHeight of steps \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h}_{s}\\)\u003c/span\u003e\u003c/span\u003e (cm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eHorizontal length of steps (cm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eStep dimension measured normal to the flow,\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}\\)\u003c/span\u003e\u003c/span\u003e (cm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{k}_{s}}_{50\\%}\\)\u003c/span\u003e\u003c/span\u003e (cm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNumber of cells\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e93,042\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e89,794\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e88,364\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe computational domain is discretized using cells of size \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Delta }x={\\Delta }y=2 mm\\)\u003c/span\u003e\u003c/span\u003e, and the associated average \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}^{+}\\)\u003c/span\u003e\u003c/span\u003e values over the steps are 48, 49, and 38, for cases A, Base and B, respectively. Boundary conditions are the same as those presented in Toro et al. (\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). The employed divergence scheme for both turbulent kinetic energy (TKE), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k\\)\u003c/span\u003e\u003c/span\u003e, and dissipation rate of TKE, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e, was the linear scheme \u003cem\u003eGauss limitedLinear\u003c/em\u003e. For the effective viscosity, the second order \u003cem\u003eGauss linear\u003c/em\u003e scheme was utilized. \u003cem\u003eGauss vannLeer\u003c/em\u003e and \u003cem\u003eGauss interfaceCompression\u003c/em\u003e were employed for the divergence terms appearing in the transport equation of the volume fraction occupied by the water. For Laplacian schemes, the \u003cem\u003eGauss linear corrected\u003c/em\u003e method was applied.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"3 Results and Discussion","content":"\u003cp\u003eThe results below correspond to those obtained for a fully developed flow, which occurs after 4 s of simulation. It was also verified that there were no regions with air close to the cavities, since only the non-aerated region of the spillway was simulated. In Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the solution of the developed flow is presented for each spillway.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Mean Flow Velocity\u003c/h2\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, the distribution of mean flow velocities at four step edges for the three stepped spillways studied, is presented. It can be clearly observed that the flow velocity increases as the flow develops along the spillway. At the closest step edge to the crest (CS1), the flow velocities are in the vicinity of 2.6 m/s for all cases, while the distribution is nearly the same, as expected, since at this point the flow is still unaware of the roughness imposed by the steps located further downstream, and the geometry in the upstream portion is the same for the three spillways, except for the short length of 12.8 cm, located immediately upstream of CS1.\u003c/p\u003e \u003cp\u003eOne can note that in all cases the maximum velocity at CS4 is close to 3.5 m/s, although minor differences among the three spillways are explained by the growth of the velocity within the boundary layer. In CS4 is observed that, at any height within the boundary layer, the stepped spillway with the smallest roughness (case B) achieves the highest velocities and the smallest boundary layer thickness, thus indicating that the flow in the water column forgets the interaction with the cavity faster when the roughness height is smaller. It is also noticeable that to comply with the principle of mass conservation, slightly higher water depths are observed in case A.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Turbulent Kinetic Energy\u003c/h2\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the normalized values of modeled turbulent kinetic energy (TKE), within the boundary layer, at four step edges for all spillways are presented. TKE values are made non-dimensional by using the corresponding maximum free-stream velocity at the step edges.\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$${k}_{norm}=k/{{V}_{max}}^{2}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{norm}\\)\u003c/span\u003e\u003c/span\u003e represents the normalized modeled TKE, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k\\)\u003c/span\u003e\u003c/span\u003e depicts the TKE, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{max}\\)\u003c/span\u003e\u003c/span\u003e is the maximum flow velocity in the water column.\u003c/p\u003e \u003cp\u003eAccording to Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the profiles of TKE for each spillway follow a clear self-similar pattern, except for case A, which is the case with the largest roughness height. This can be attributed to the roughness layer covering the entire flow depth and not a portion of it. This is particularly relevant since it provides evidence of the existence of a maximum roughness height for which the distribution of TKE can be expressed through a unique curve. In other words, this would indicate the existence of an upper limit for the size of the cavity (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}\\)\u003c/span\u003e\u003c/span\u003e) for which distributions of TKE at step edges can be collapsed into one single curve. It can also be observed that values of TKE are larger in the stepped spillway with larger roughness height, reaching maximum values ranging between 0.033 and 0.037, which are located at a height between 10 and 15% of the thickness of the boundary layer. For cases Base and B, maximum values of TKE are approximately 0.028 and 0.023 respectively, and they are both located at an approximate height of 15% of the thickness of the boundary layer.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, the TKE values in the water column at the middle of the steps, corresponding to a roughness height of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{k}_{s}}_{50\\%}\\)\u003c/span\u003e\u003c/span\u003e, are presented. Remarkably, just as the distributions of TKE at step edges (previous Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), cases B and Base present TKE values that can be reasonably well grouped by means of a single curve. This is different from case A, which presents greater variability in TKE values, especially at relative water depths between 0.5 and 0.8; this is, from just below the pseudobottom to above the pseudobottom. Inside the cavity, the values are well grouped for case A. Maximum values of TKE are located above the pseudobottom with representative values of 0.019, 0.022 and 0.027 for cases B, Base and A, respectively.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Boundary layer thickness \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varvec{\\delta }\\)\u003c/span\u003e\u003c/span\u003e and relative roughness (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varvec{k}}_{\\varvec{s}}/\\varvec{h}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/h2\u003e \u003cp\u003eThe growth of the boundary layer thickness (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003e) and the distribution of relative roughness (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}/h\\)\u003c/span\u003e\u003c/span\u003e) for the three spillways and four cross sections are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Noteworthy is the similar rate of growth of the boundary layer for the three stepped spillways between cross sections \u003cspan refid=\"Sec1\" class=\"InternalRef\"\u003e1\u003c/span\u003e to \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e4\u003c/span\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea). This corresponds to the approximate slope of the line connecting the thickness of the boundary layer in those four points. However, in cross section \u003cspan refid=\"Sec1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, for both cases, Base and B, the boundary layer thickness is approximately 1.6 cm, whereas in case A the value is larger and equal to 2.4 cm. This is because in case A the flow recognizes more quickly the existence of a larger roughness height.\u003c/p\u003e \u003cp\u003eIt can be inferred from Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb that only in cases B and Base, water depths obtained at step edges are a fraction of the size of the roughness element (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}/h\\)\u003c/span\u003e\u003c/span\u003e smaller than ~\u0026thinsp;0.8). This value will be considered as the upper limit to obtain self-similar distributions of flow variables.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Rate of dissipation of turbulent kinetic energy\u003c/h2\u003e \u003cp\u003eThe normalized rate of dissipation of turbulent kinetic energy (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\epsilon }_{norm}\\)\u003c/span\u003e\u003c/span\u003e) can be described as:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$${\\epsilon }_{norm}=\\frac{\\epsilon \\delta }{{V}_{max}^{3}} \\left(1\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e is the rate of dissipation of TKE and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003e is the boundary layer thickness. The \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\epsilon }_{norm}\\)\u003c/span\u003e\u003c/span\u003e values computed from the three simulated steeped spillways at the selected four cross sections are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIt is very clear that the peak rate of dissipation of energy occurs in proximity to the step edges at each selected cross section, progressively decreasing as distance from the pseudobottom increases. For all cases (Base, A and B) and cross sections (CS1-CS4) the curves of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e describe a pseudo self-similar shape. This is because a self-similar distribution of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e values is observed only for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(y/\\delta \u0026gt;0.2\\)\u003c/span\u003e\u003c/span\u003e. Indeed, it is evident that close to the step edges (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(y/\\delta \u0026lt;0.2\\)\u003c/span\u003e\u003c/span\u003e), there is a range of maximum values of epsilon and therefore a unique curve cannot be employed to summarize those values. Perhaps it is worth emphasizing here that the rate of dissipation of turbulent kinetic energy is a highly challenging variable to model, as evident from its very definition. Strictly speaking, only a characterization of resolved turbulent flow fluctuations would enable the estimation of this variable. Therefore, what is described here pertains more to the mathematical behavior of the flow solution near the pseudobottom.\u003c/p\u003e \u003cp\u003eFor the spillway with smaller steps, case B, the self-similarity is more evident for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(y/\\delta \u0026gt;0.05\\)\u003c/span\u003e\u003c/span\u003e. In that case, epsilon values are well grouped, and therefore the rate of dissipation of TKE can be described by a single curve, independent of the step (cross section). Since case B presents the smaller roughness, its behavior better resembles the flow of conventional open channel flows. The maximum values of modeled rates of dissipation for case B are found right at the pseudobottom, with values ranging from 0.015 (CS1) to 0.017 (CS4), which indicates that the rate of dissipation is nearly constant at the pseudobottom along the spillway. Slightly larger differences can be found at the pseudobottom for Base case, where rates of dissipation range from 0.018 (CS1) to 0.026 (CS4). However, up, and not too far from the steps, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(y/\\delta \u0026gt;0.10\\)\u003c/span\u003e\u003c/span\u003e, the behavior of the rate of dissipation becomes independent of the cross section, and therefore self-similar. Finally, for the larger roughness height, case A, self-similarity of the rate of dissipation (independent of the location of the step) occurs at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(y/\\delta \\sim0.2\\)\u003c/span\u003e\u003c/span\u003e and above, since below this height, the values of the rate of dissipation become dependent of the flow development as the fluid moves down the stepped spillway. The rate of dissipation values at the pseudobottom range from 0.025 (CS1) to 0.034 (CS4) for case A. From the above discussion it is possible to recognize that the height at which the rate of dissipation of TKE starts to be self-similar is directly correlated to the roughness height, since the larger the size of the cavity the larger the height at which self-similar behavior starts to occur. In a prior study conducted by Toro et al. (\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), the presence of self-similarity in turbulent kinetic energy (TKE) and the rate of dissipation of TKE for the Base case was demonstrated. However, it is important to remark that this investigation did not consider variations in roughness heights.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Mean pressure\u003c/h2\u003e \u003cp\u003eDimensionless pressures (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}^{*}\\)\u003c/span\u003e\u003c/span\u003e) at step edges and within the cavities are plotted in Figs.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, respectively. Pressures (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(p\\)\u003c/span\u003e\u003c/span\u003e) were made dimensionless by using the projection of the water depth on the vertical and they were calculated as:\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$${p}^{*}=\\frac{p}{\\gamma hcos\\left(\\theta \\right)} \\left(1\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e is the specific weight of water (N/m\u003csup\u003e3\u003c/sup\u003e), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(h\\)\u003c/span\u003e\u003c/span\u003e is the flow depth measured normal to the spillway pseudobottom, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e is the slope of the stepped spillway. Since the denominator in the above equation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma hcos\\left(\\theta \\right)\\)\u003c/span\u003e\u003c/span\u003e, represents the hydrostatic pressure, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}^{*}\\)\u003c/span\u003e\u003c/span\u003e values above, equal, and below one, indicate pressures over, equal to or less than the hydrostatic pressure, respectively.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e it is shown that from the water surface down to about 40% of the dimensionless water depth, case B has a pressure distribution at step edges which is remarkably like the hydrostatic pressure distribution, attributed to the small cavity. Hence, the pressure field is affected by the influence of the cavity only in the first 40% of the water column, reaching maximum values varying from 1.5 to 2. For cases A and Base, pressure distributions at step edges remain very distant from the hydrostatic distribution. This is especially true for case A, due to the larger roughness height. In case A, the highest-pressure values are located approximately at 20% of the water depth, with a range of values between 1.4 (CS1) and 2.2 (CS2). Finally, for Base case, maximum pressures are located close to step edges as in case B, with values varying from 1.5 to 2.5.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, the dimensionless pressure inside the step cavities and extending up to a distance of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003e above the pseudobottom is depicted for various cross sections. Pressure values are thus analyzed from the cavity wall (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Y=0\\)\u003c/span\u003e\u003c/span\u003e) up to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Y=K{s}_{50\\%}+\\delta\\)\u003c/span\u003e\u003c/span\u003e (see Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) since the focus is on the relationship between the flow in the cavity and the flow in the shear layer, just above the pseudobottom.\u003c/p\u003e \u003cp\u003eCase B, with the smallest roughness height, is the only case in which pressure values inside the cavity are larger than those obtained from the hydrostatic distribution, for all cross sections studied. Pressure profiles resemble an S-shaped curve where two inflection points can be identified: one at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}^{*}\\sim0.08\\)\u003c/span\u003e\u003c/span\u003e and another at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}^{*}\\sim0.30\\)\u003c/span\u003e\u003c/span\u003e. For \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}^{*}\u0026gt;0.40\\)\u003c/span\u003e\u003c/span\u003e, and moving towards the free surface, all curves follow a quasi-linear distribution.\u003c/p\u003e \u003cp\u003eOn the contrary, cases A and Base, with larger roughness heights, exhibit pressures below the hydrostatic pressure, attributed to the recirculating flow inside the cavity. This provides evidence that the pressure distribution within the cavity ceases to be hydrostatic once a particular roughness height is exceeded. Below the roughness size threshold, and due to the lack or minimum recirculation within the cavity, the pressure profile is always larger than hydrostatic. As the cavity becomes larger, flow recirculation grows, decreasing pressure below the hydrostatic pressure.\u003c/p\u003e \u003cp\u003eFor the Base case two inflection points can be identified: the first one is located near \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}^{*}\\sim0.30\\)\u003c/span\u003e\u003c/span\u003e, where minimum pressure values (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}^{*}\\)\u003c/span\u003e\u003c/span\u003e) range from \u0026minus;\u0026thinsp;0.8 (CS4) up to -0.35 (CS1). The second inflection point is around \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}^{*}\\sim0.85\\)\u003c/span\u003e\u003c/span\u003e. In case A with the largest cavity, on the other hand, only one inflection point is observed, close to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}^{*}\\sim0.40\\)\u003c/span\u003e\u003c/span\u003e, where the pressure reaches its minimum value for all four cross sections. It is also observed that the initial two cross sections CS1 and CS2, exhibit a behavior that closely resembles a single curve with a minimum pressure value around \u0026minus;\u0026thinsp;0.25. This drastically changes for profiles in cross sections CS3 and CS4, where minimum pressures move to about \u0026minus;\u0026thinsp;0.55 and \u0026minus;\u0026thinsp;0.7, respectively.\u003c/p\u003e \u003cp\u003eCases A and Base share the feature of pressure distributions lower than the hydrostatic counterpart. They also share that further downstream, the minimum pressure values within the cavity are progressively decreasing. For the larger roughness height (case A), however, the pressure is distributed more smoothly than in Base case, which presents a more pronounced pressure peak at the lowest inflection point.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003eThe present work corresponds to a numerical study of the flow over steep stepped spillways, with the emphasis placed on the effect of the roughness height on the distribution of the mean flow velocity and pressure, and distribution of modeled turbulence statistics, namely the turbulent kinetic energy (TKE) and rate of viscous dissipation (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e). To that end, three cases were analyzed. Case A, with a step height of 10 cm and a horizontal length of 8 cm (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}\\)\u003c/span\u003e\u003c/span\u003e=6.25 cm) which corresponds to the larger cavity dimensions; Base case with half the dimensions of case A (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}\\)\u003c/span\u003e\u003c/span\u003e=3.12 cm) and case B, with half the dimensions of Base case (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}\\)\u003c/span\u003e\u003c/span\u003e=1.56 cm).\u003c/p\u003e \u003cp\u003eFrom the obtained results, the following conclusions can be extracted:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eFor Cases B and Base, which are the stepped spillways with the smaller steps, distribution of turbulent kinetic energy within the boundary layer at the step edges can be summarized in one single curve. This self-similarity behavior is not observed for Case A, which has the larger steps. According to this, it might be argued that self-similarity for the modeled values of TKE is met only for values \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}/h\u0026lt;0.8\\)\u003c/span\u003e\u003c/span\u003e which correspond to the upper limit for cases B and Base analyzed in this study.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe greater the size of the cavity, the greater the maximum velocities inside the cavities. Maximum velocities inside the cavities for cases A, Base and B are: 0,89; 0.76; y 0.28 m/s, respectively.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eFor a given location, it was found that the larger the roughness height is, the larger the thickness of the boundary layer associated. For example, at cross section CS4, boundary layer thicknesses of 2.8, 2.4 and 2.1 cm were obtained for cases A, Base and B, respectively. However, the rate of growth of the boundary layer for case A with greater steps is like that of case B with the smaller steps.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eRegardless of the size of the step cavity, maximum velocities in the water column are nearly the same.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe velocity distribution at step edges for the three stepped spillways can be collapsed in one single curve. The flow velocity at the first control step is 2.6 m/s and it grows downstream up to 3.6 m/s at the fourth control step.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eMaximum values of TKE at step edges are located at 12% of the boundary layer thickness with a representative non-dimensional value of 0.037.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ePressure distributions at step edges follow a distribution greater than the hydrostatic for the three studied cases. For case A, with greater steps, maximum values of pressure are located at 20% of the boundary layer, whereas in cases B and Base, maximum values for pressure are located close to step edges.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e"},{"header":"Statements and Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe first author gratefully acknowledges financial support of the National Fund for Scientific and Technological Development (FONDECYT) through ANID Iniciaci\u0026oacute;n en Investigaci\u0026oacute;n Project N\u0026deg; 11221196.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no financial or non-financial interests to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eJuan Pablo Toro contributed to the study conception and design. Implementation of the numerical simulations and analysis were performed by Juan Pablo Toro and Alex Blanc. Figures preparation and data collection was performed by Alex Blanc and Sebasti\u0026aacute;n Sep\u0026uacute;lveda. The first draft of the manuscript was written by Juan Pablo Toro. Patricio Moreno-Casas contributed to the manuscript design and commented on previous versions of the manuscript. All authors read and approved the final manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAmador A (2005) Comportamiento hidr\u0026aacute;ulico de los aliviaderos escalonados en presas de hormig\u0026oacute;n compactado. Ph.D. thesis, Technical University of Catalonia (UPC), Barcelona, Spain (in Spanish)\u003c/li\u003e\n\u003cli\u003eAmador A, S\u0026aacute;nchez-Juny M, Dolz J (2009) Developing flow region and pressure fluctuations on steeply sloping stepped spillways. J Hydraul Eng:1092\u0026ndash;1100. https://doi.org/0.1061/(ASCE)HY.1943-7900.0000118\u003c/li\u003e\n\u003cli\u003eBarenblatt GI, Isaakovich BG (1996) Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics (No. 14). Cambridge University Press. https://doi.org/10.1017/CBO9781107050242\u003c/li\u003e\n\u003cli\u003eBaroud CN, Plapp BB, She ZS, Swinney HL (2002) Anomalous self-similarity in a turbulent rapidly rotating fluid. Physical Review Letters 88:114501. https://doi.org/10.1103/PhysRevLett.88.114501\u003c/li\u003e\n\u003cli\u003eBoes R, Hager W (2003) Two-phase flow characteristics of stepped spillways. J Hydraul Eng 129:661-670. https://doi.org/10.1061/(ASCE)0733-9429(2003)129:9(661)\u003c/li\u003e\n\u003cli\u003eCenedese C, Linden PF (2014) Entrainment in two coalescing axisymmetric turbulent plumes. 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Water Resour Manage 34:87\u0026ndash;100. https://doi.org/10.1007/s11269-019-02395-5\u003c/li\u003e\n\u003cli\u003eErcan A, Kavvas ML, Haltas I (2014) Scaling and self‐similarity in one‐dimensional unsteady open channel flow. Hydrological Processes 28:2721-2737. https://doi.org/10.1002/hyp.9822\u003c/li\u003e\n\u003cli\u003eForoudi A, Roushangar K, Saneie M et al (2022) Evaluating the Effect of Downstream Channel Width Variation on Hydraulic Performance of Arched Plan Stepped Spillways. Water Resour Manage 36:4237\u0026ndash;4253. https://doi.org/10.1007/s11269-022-03250-w\u003c/li\u003e\n\u003cli\u003eGiudicianni C, Di Nardo A, Greco R et al (2021) A Community-Structure-Based Method for Estimating the Fractal Dimension, and its Application to Water Networks for the Assessment of Vulnerability to Disasters. Water Resour Manage 35:1197\u0026ndash;1210. https://doi.org/10.1007/s11269-021-02773-y\u003c/li\u003e\n\u003cli\u003eGonzalez CA, Takahashi M, Chanson H (2008) An experimental study of effects of step roughness in skimming flows on stepped chutes. Journal of Hydraulic Research 46(sup1):24-35. https://doi.org/10.1080/00221686.2008.9521937\u003c/li\u003e\n\u003cli\u003eGuinot V, Delenne C, Soares-Fraz\u0026atilde;o S (2023) Self-similar solutions of shallow water equations with porosity. Journal of Hydraulic Research 61:109-119. https://doi.org/10.1080/00221686.2022.2106598\u003c/li\u003e\n\u003cli\u003eHassan MK, Hassan MZ, Pavel NI (2011) Dynamic scaling, data-collapse and self-similarity in Barab\u0026aacute;si\u0026ndash;Albert networks. Journal of Physics A: Mathematical and Theoretical 44:175101. https://doi.org/10.1088/1751-8113/44/17/175101\u003c/li\u003e\n\u003cli\u003eHeller V (2017) Self-similarity and Reynolds number invariance in Froude modelling. Journal of Hydraulic Research 55:293-309. https://doi.org/10.1080/00221686.2016.1250832\u003c/li\u003e\n\u003cli\u003eHellstr\u0026ouml;m LH, Marusic I, Smits AJ (2016) Self-similarity of the large-scale motions in turbulent pipe flow. Journal of Fluid Mechanics 792:R1. https://doi.org/10.1017/jfm.2016.100\u003c/li\u003e\n\u003cli\u003eKartal V, Emiroglu ME (2023) Hydraulic Performance of Sharp-Crested Side Slit Weirs. Water Resour Manage 37:1297\u0026ndash;1319. https://doi.org/10.1007/s11269-023-03433-z\u003c/li\u003e\n\u003cli\u003eKuhn P, Soria J, Oberleithner K (2021) Linear modelling of self-similar jet turbulence. Journal of Fluid Mechanics 919:A7. https://doi.org/10.1017/jfm.2021.292\u003c/li\u003e\n\u003cli\u003eLin C, Hsieh SC, Lin IJ, Chang KA, Raikar RV (2012) Flow property and self-similarity in steady hydraulic jumps. Experiments in Fluids 53:1591-1616. https://doi.org/10.1007/s00348-012-1377-2\u003c/li\u003e\n\u003cli\u003eMatos J (1999) Air entrainment and energy dissipation in flow over stepped spillways. Ph.D. thesis, IST, Lisbon, Portugal (in Portuguese)\u003c/li\u003e\n\u003cli\u003eMeireles I, Renna F, Matos J, Bombardelli FA (2012) Skimming, nonaerated flow on stepped spillways over roller compacted concrete dams. J Hydraul Eng 10.1061/(ASCE)HY.1943-7900.0000591:870\u0026ndash;877. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000591\u003c/li\u003e\n\u003cli\u003eRajaratnam N (1990) Skimming flow in stepped spillways. Journal of Hydraulic Engineering 116:587-591. https://doi.org/10.1061/(ASCE)0733-9429(1990)116:4(587)\u003c/li\u003e\n\u003cli\u003eS\u0026aacute;nchez-Juny M (2001) Comportamiento hidr\u0026aacute;ulico de los aliviaderos escalonados en presas de hormig\u0026oacute;n compactado. An\u0026aacute;lisis del campo de presiones. Ph.D. thesis, Technical Univ. of Catalonia (UPC), Barcelona, Spain (in Spanish)\u003c/li\u003e\n\u003cli\u003eToro JP, Bombardelli FA, Paik J, Meireles I, Amador A (2016) Characterization of turbulence statistics on the non-aerated skimming flow over stepped spillways: a numerical study. Environmental Fluid Mechanics 16:1195\u0026ndash;1221. https://doi.org/10.1007/s10652-016-9472-1\u003c/li\u003e\n\u003cli\u003eToro JP, Bombardelli FA, Paik J (2017) Detached eddy simulation of the nonaerated skimming flow over a stepped spillway. Journal of Hydraulic Engineering 143:04017032. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001322\u003c/li\u003e\n\u003cli\u003eWang Q, Qu F, Sun D, Bai J (2023) Numerical study of instabilities and compressibility effects on supersonic jet over a convex wall. Journal of Fluid Mechanics 954:A6. https://doi.org/10.1017/jfm.2022.977\u003c/li\u003e\n\u003cli\u003eWood IR (1991) Free surface air entrainment on spillways. In: Air entrainment in free-surface flows. A.A. Balkema, Rotterdam, The Netherlands, pp 55-84\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Stepped spillway, Self-similarity, Roughness height, Boundary layer, Turbulent kinetic energy","lastPublishedDoi":"10.21203/rs.3.rs-3538512/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3538512/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe stepped spillway design is widely used globally for its energy dissipation capabilities through its stepped design and air entrainment capacity, crucial for safe hydraulic structure operation. The most common condition found in stepped spillways is the skimming flow regime. This study numerically investigated the influence of the roughness height (K\u003csub\u003eS\u003c/sub\u003e) on the distribution of mean flow variables and modeled turbulence statistics in the non-aerated portion of a steep stepped spillway with a constant angle of 51.34\u0026deg;. Three roughness heights of 6.26, 3.13 and 1.57 cm were considered corresponding to a relation step height/horizontal length of 10/8, 5/4 and 2.5/2 (cm/cm), respectively. The results show that profiles of modeled turbulent kinetic energy (TKE) at different step edges can be collapsed into one single curve for relative roughness (Ks/h) smaller than 0.8. For larger ratios of roughness height to water depth, this behavior is not observed and for a given distance away from the wall, values of TKE show a broad range of values, with increasing values as boundary layer grows. A similar trend was obtained for the distribution of TKE along a water column that includes the cavity. The existence of an upper limit for the ratio between the roughness height and the water depth can now be incorporated in the hydraulic design of new steep stepped spillways.\u003c/p\u003e","manuscriptTitle":"Influence of roughness height on the distribution of modeled turbulence statistics in non-aerated skimming flows in steep stepped spillways","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-11-07 13:46:03","doi":"10.21203/rs.3.rs-3538512/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"304909cf-13a3-435b-9dec-5955ba71546e","owner":[],"postedDate":"November 7th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-08-05T00:50:22+00:00","versionOfRecord":[],"versionCreatedAt":"2023-11-07 13:46:03","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3538512","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3538512","identity":"rs-3538512","version":["v1"]},"buildId":"rHA-KDH7Qsr4HCuvH75dn","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}