NUMERICAL STUDY OF A TURBULENT HYDRAULIC JUMP

Transcription

1 NUMERICAL STUDY OF A TURBULENT HYDRAULIC JUMP Qun Zhao Shubhra K. Misra Ib A. Svendsen (Member, ASCE) and James T. Kirby (Member, ASCE) ABSTRACT In this paper, we describe the numerical simulation of a turbulent hydraulic jump. The numerical model is based on RIPPLE (Kothe et al., 994) with two turbulence closure submodels: the standard k model of Launder and Spalding (974) and a multi-scale k l model of Zhao et al. (24). Model results are compared to LDV data of Bakunin (995) and Svendsen et al. (2). Keywords: hydraulic jump, bore, spilling breaker INTRODUCTION Hydraulic jumps and bores are similar flows even though the hydraulic jump is stationary and the bore is propagating. In fact a bore can be viewed and analyzed in a coordinate system moving at the bore propagation speed as a stationary hydraulic jump. Hydraulic jumps are commonly used as energy dissipators and they have been studied intensively by hydraulic engineers mainly through laboratory experiments. Hydraulic jumps and bores are also of interest to coastal engineers due to the similarity with broken waves inside the surf zone (Peregrine and Svendsen, 978, Madsen and Svendsen, 983). Spilling breakers are often modeled as a hydraulic jump with a stagnant eddy (roller) riding on the turbulent front face (Svendsen and Madsen, 984). Most of the studies focusing on hydraulic jumps are however based on laboratory experiments and semi-analytical solutions (Svendsen et al., 2). The numerical studies reported so far are mainly based on nonlinear shallow water equations (NSWE), where jumps and bores are considered as a discontinuity (shocks) of the water surface. An early study by Hibberd and Peregrine (975) used a LaxWendroff scheme where the dissipation was implicitly represented in the numerical scheme. More advanced shock capture methods such as Weighted Averaged Flux (WAF) method are used by recent NSWE solvers such as Brocchini et al.(2) and Hubbard and Dodd (22). However, as pointed out by Madsen and Svendsen (983), these methods based on NSWE are unable to correctly describe the shape of the free surface and the velocities underneath it. In the present work, we apply a Navier-Stokes solver based on the VOF method (Kothe et al., 994, Zhao et. al 24) to study a turbulent hydraulic jump. We will focus on the mean flow motions in the hydraulic jump. Another aspect that distinguishes the present study from others (such as Chippada et al., 994; Liu and Drewes, 995; Ma et al., 22) is that in this Center for Applied Coastal Research, University of Delaware, Newark, DE 976, USA. Corresponding author zhao@coastal.udel.edu, fax:

2 work we will focus on weak hydraulic jumps/bores, i.e., Froude number less than 2, due to their similarity to the bores observed in the natural beach. At the first stage of this work, we wish to validate our numerical model using the laboratory measurements of Bakunin (995) and Svendsen et al. (2). GOVERNING EQUATIONS The numerical model used here is based on the two-dimensional VOF model RIPPLE (Kothe et al., 994). The governing equations are the continuity and momentum equation for incompressible flow in the x z plane, u j = () u i t + u u i j = p + ν 2 u i + g i, (2) ρ x i where i =, 2 are indices in the x and z direction, respectively, and j is a repeated dummy index. ν is the kinematic viscosity and g i the gravitational acceleration, where g =. p is the pressure, and ρ the fluid density with t the time. The pressure is solved using an incomplete Cholesky conjugate gradient technique for the pressure Poisson equation. The VOF method uses a volume of fluid function F (x, z, t) to to track the motion of the free surface. A unit value of F corresponds to a cell full of fluid, while a zero value indicates that the cell contains no fluid. Cells with F values between zero and one contain a free surface. The time evolution of F is governed by, F t + u F j =. (3) The instantaneous equations ()-(3) cannot be solved directly due to the high Reynolds number and the limitation of the grid size. The governing equations are thus averaged over time or space. Upon using the Boussinesq eddy viscosity assumption, the mean momentum equations read ū i t + ū ū i j = p + [(ν + ν T ) ū i ] + g i. (4) ρ x i where ν T is the eddy viscosity. The variables with an overbar denote the mean variables either time-averaged or space-averaged depending on the turbulence model used. TURBULENCE MODELING The k ɛ model The standard k ɛ model (Launder and Spalding, 974; Rodi, 98) is given by K t + u K j = (ν + ν T K ) + P ROD ɛ. σ K (5) ɛ t + u ɛ j = (ν + ν T ɛ ) + ɛ(c ɛ P ROD c ɛ2 ɛ). (6) σ ɛ where K and ɛ are the turbulent kinetic energy and dissipation rate. P ROD denotes the production term due to the mean flow [ ( ūi P ROD = ν T + ū )( j ūi + ū ] ) j, (7) 2 x i x i 2

3 K PROD ε ε2 ε 3 ε 4 ε N K K 2 K 3 K 4 ε N K N FIG.. κ κ 2 κ3 κ κ N κ N 4 Partition of the spectra density κ and the eddy viscosity relates the kinetic energy and dissipation rate as ν T = c µ K 2 We use the standard coefficients here: c u =.9, σ K =., σ ɛ =.3, c ɛ2 =.92, c ɛ = c ɛ2 κ 2 /σ ɛ c 2 µ. The Multi-scale Turbulence Model As for the multi-scale turbulence model, we follow the work of Zhao et al. (24) to set up N levels of k l equations, in which turbulent kinetic energy can be transferred from larger length scales to smaller ones explicitly (see Figure ). In this approach, the production term at the first level is dominated by the large eddies, i.e., the smallest resolved scale (the grid scale, in this case), [ ( ūi P ROD = ν t + ū )( j ūi + ū ] ) j, (9) 2 x i x i The dissipation term is the k l type dissipation, ɛ ɛ n = C d k 3 2 n /l n, n < N, () where the length scale for each partition is x z l n = 2 n n < N () Within each partition n >, production is set equal to dissipation at the next n-level grid scale partition, P ROD n = ɛ n 2 n N. (2) Then the total sub-grid scale (SGS) turbulent kinetic energy is the sum of the kinetic energy in all partitions, k = k + k k N = Σ N n=k n. (3) 3 (8)

4 We also assume that turbulent kinetic energy leaves the system at the smallest scales, therefore the total dissipation of the system is characterized as and Then the nth level k l equation reads, k n t + u k n j = ɛ = ɛ N. (4) ( ν t n σ k k n ) + P ROD n ɛ n. (5) C d =.2, C s =. and σ k =. are used in the present study. The eddy viscosity ν tn at the nth level is set to all the scales smaller than n, ν tn = Σ N i=n C sk 2 i l i n < N. (6) ν T = ν t, (7) in (4). For the results presented below, N = 3, i.e., three levels of k l equations are solved. NUMERICAL CONSIDERATIONS The numerical scheme in this work follows RIPPLE (Kothe et al., 994) and Zhao et al. (24). At the bottom boundary, we use the wall boundary condition (Launder and Spalding, 974) to match the near wall velocity to the wall function. At inflow, we use the power law velocity profile to approximate the velocity at the first measured point in the experiment. Downstream boundary condition is forced to conserve the volume flux. PRELIMINARY RESULTS: THE MEAN QUANTITIES.5 z/h x/h FIG. 2. Vector plot and surface elevation for the hydraulic jump. The Froude number of this case is F r =.46. Circles are data from Svendsen et al. (2). The computational results are obtained on a grid δx =.cm, δy =.25cm with the multi-scale turbulence model. Model results are averaged over 4 seconds. The numerical model is tested using the hydraulic jump data of Bakunin (995) and Svendsen et al. (2). The preliminary results using the VOF code and the multi-scale turbulence model are shown in Figures 2 and 3. In general, the model agrees with the measurements reasonably well, but underestimates the horizontal velocities downstream of the jump. Figures 4 presents the computed mean vorticity field showing positive vorticity at the roller area and negative vorticity near the bottom. 4

5 gauge gauge2 gauge3 gauge4 gauge z/h gauge6 gauge7 gauge8 gauge z/h FIG. 3. Comparison of measured (circles, Svendsen et al., 2) and modeled (line, the multi-scale turbulence model) mean horizontal velocity profiles. Gauge locations correspond to the circles in Figure 2. SUMMARY The preliminary results show that the numerical model agrees reasonably well with the lab measurements for the mean flow. But the model underestimates the horizontal velocities downstream of the jump. We are currently working on comparing the turbulent motions with the measurements and hope to present those results at the conference. ACKNOWLEDGMENTS This work is supported by the National Oceanographic Partnership Program (NOPP), grant N REFERENCES Bakunin, J. (995): Experimental study of hydraulic jumps in low Froude number range. MCE thesis, Department of Civil AND Environmental Engineering, University of Delaware, Newark, DE 97. Brocchini, M., R. Bernetti, A. Mancinelli and G. Albertini (2): An efficient solver for nearshore flows based on WAF method. Coastal Engineering, vol 43, pp Chippada, S., B. Ramaswamy and M. F. Wheeler (994): Numerical simulation of hydraulic jump. International Journal of Numerical Methods in Engineering. vol. 37, pp. 5

6 FIG. 4. Computed mean vorticity field showing positive vorticity in the roller area and negative vorticity at the bottom. The unit is /s Hibberd, S. and D. H. Peregrine (979): Surf and run-up on a beach: a uniform bore. Journal of Fluid Mechanics. vol. 95, part 2, pp Hubbard, M. and N. Dodd (22): A 2D numerical model of wave run-up and overtopping. Coastal Engineering, vol. 47: () -26. Kothe, D. B., R.C. Mjolsness and M. D. Torrey (994): RIPPLE: A computer program for incompressible flows with free surfaces. Los Alamos Report, LA-727-MS. Launder B. E. and D. B. Spalding (974): The numerical computation of turbulent flows. Computational Methods in Applied Mechanics and Engineering. vol. 3, pp Liu, Q. C. and U. Drews (995): Turbulence characteristic es in free and forced hydraulic jumps. Journal of Hydraulic Research, vol. 32, 994, pp Ma, F., Y. Hou and P. Prinos (22): Numerical calculation of submerged hydraulic jump. Journal of Hydraulic Research. vol. 39, No. 5, pp. -. Madsen, P. A. and I. A. Svendsen (983): Turbulence bores and hydraulic jumps. Journal of Fluid Mechanics. vol. 29, pp.-25. Peregrine, D. H. and I. A. Svendsen (978): Spilling breakers, bores and hydraulic jumps. Proc. 6th ICCE, Svendsen, I. A. and P. A. Madsen (984): A turbulent bore on a beach. Journal of Fluid Mechanics. vol. 48, pp

7 Svendsen, I. A., Veeramony, J., Bakunin, J. and J. T. Kirby (2): The flow in weak turbulent hydraulic jump. Journal of fluid Mechanics. vol 48, pp Zhao, Q., S. Armfield and K. Tanimoto (24) : Numerical simulation of breaking waves by a multi-scale turbulence model.coastal Engineering 5, pp