Numerical modeling of complex turbulent flows

Transcription

1 ISTP-16, 5, PRAGUE 16 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA Numerical modeling of complex turbulent flows Karel Kozel Petr Louda Jaromír Příhoda Dept. of Technical Mathematics CTU Prague, Karlovo nám. 13, CZ Prague Institute of Thermomechanics AS CR, Dolejškova 5, CZ-18 Prague 8 Corresponding author: P. Louda, louda@marian.fsik.cvut.cz Keywords: jet in cross-flow, impinging jet, artificial compressibility, model,, heat transfer Abstract The work deals with the solution of two cases of turbulent incompressible flows: impinging jet flow and round jet in cross-flow. The flows are considered turbulent and statistically steady. The mathematical model is based on Reynolds averaged Navier-Stokes equations with a twoequation turbulence model. The turbulence models are low Reynolds number eddy-viscosity one an model. The system of equations is solved by artificial compressibility method. The discretisation uses finite volume method with higher order upwind approximation for convective term and implicit discretisation in time. For both cases results are compared with experimental data. In the case of impinging jet flow, the heat transfer from impingement wall is also solved. 1 Introduction In this work, we investigate two cases of 3D turbulent flow using the Reynolds averaging approach. The first one is round jet in cross-flow, the second one impinging jet flow. Using sufficiently accurate implicit finite volume method, we study the inluence of turbulence modeling on the results. We use one eddy viscosity and explicit algebraic Reynolds stress model for turbulent momentum trabsfer, and three different turbulent heat flux clusores. Mathematical model The turbulent flow-field in both cases is assumed to be statistically steady. Then the mean flowfield is governed by Reynolds averaged Navier- Stokes (RANS) equations in Cartesian coordinates x i u i t + u ju i ui x i =, = p + (νs ij τ ij ), x i S ij = 1 ( ui + u ) j, (1) x i where u i is mean velocity vector, p kinematic pressure, ν laminar kinematic viscosity, and τ ij Reynolds stress tensor. We apply two closures for τ ij : the (extended) eddy viscosity one of turbulence model and explicit algebraic Reynolds stress model () due to Wallin, Hellsten. The Reynolds stress by model reads τ ij = ν T S ij + 3 kδ ij, ν T = a 1 k max(a 1 ω, ΩF ), a 1 =.31, () where k is turbulent kinetic energy, ω specific dissipation rate, Ω magnitude of vorticity, and F a switching function given in [6]. The 1

2 constitutive relation is given in terms of anisotropy tensor a ij [11, ] τ ij = a ij k + 3 kδ ij, (3) a ij = β 1 τ T S ij + + β 3 τ (Ω ik Ω kj II Ω δ ij /3) + + β 4 τ (S ik Ω kj Ω ik S kj ) + + β 6 τ 3 (S ik Ω kl Ω lj + Ω ik Ω kl S lj IV δ ij /3) + + β 9 τ 4 (Ω ik S kl Ω lm Ω mj Ω ik Ω kl S lm Ω mj ), where τ is turbulent time scale limited from below by Kolmogorov time scale ν/ɛ τ = max Ω ij is rotation rate tensor ( ) k ν ɛ, 6, ɛ =.9kω, (4) ɛ Ω ij = 1 ( ui u ) j, (5) x i and II Ω, IV are invariants formed from S ij, Ω ij given in [] as well as the coefficients β. Both turbulence models use k-ω model equations of similar form Dk Dt = P β kω + [ Dω Dt = γ ω k P βω + + [ (ν + σ ω ν T ) ω ] (ν + σ k ν T ) k + σ d ω ], k ω, (6) where D /Dt / t + (u j )/, and coefficients for model are given in [6]. The efective eddy-viscosity in is given by [] ν T = C µ kτ, C µ = 1 (β 1 + II Ω β 6 ). (7) For the remaining coefficients see also []. 3 Numerical method The system of RANS and turbulence model equations is solved by an artificial compressibility method, which consists of adding a pressure time derivative into the continuity equation: 1 β p t + u i x i =, (8) where the parameter β has been chosen equal to the maximum velocity in the solution domain. The steady solution of the original system is obtained for t. To discretize the equations in space, the cellcentered finite volume method is used. The grid consists of hexahedral cells and is structured and multi-block. An example of handling circular nozzle in the case of jet in cross-flow is shown in Fig. 1. The convective terms are discretized using 3rd order van Leer upwind interpolation (without a limiter). The viscous terms are approximated using nd order central scheme, with cell face derivatives computed on a dual grid of octahedrons constructed over each face of primary grid. Pressure stabilization in the form of pressure Laplacian is added to the continuity equation to prevent pressure-velocity decoupling. The time integration uses backward Euler (implicit) scheme. The Newton linearized implicit operator contains also convective terms, however has the 1st order upwind stencil. In, linear part of turbulent diffusion corresponding to eddy viscosity (7) is discretized implicitely, the remaining part explicitely. Only the negative part of source terms in the turbulence model is discretized implicitly. The resulting block 7-diagonal system of algebraic equations is solved using block relaxation method with direct tri-diagonal solver for slected family of grid lines [3]. The CFL numbers range between 3 and 1. 4 Jet in cross flow This section deals with the simulation of flowfield generated by a round jet injected into a cross-flow of same fluid. The case modeled is given by the available measurement [1], where the speeds were 4.8 m/s and 1. m/s in the jet nozzle and in the cross-flow, respectively, with

3 Fig. 1: Detail of the grid in plane (x, y) z x B H y d Uj A Ue C C/ Fig. : Computational domain for jet in cross-flow no effects of compressibility. The turbulence intensity in both inlets was very low (under.%). The rectangular computational domain approximating aerodynamic tunnel test section is shown in Fig.. The dimensions are (d, A, B, C, H) = (1, 4, 4, 19, 14.8) d, where d = 13.3 mm, and the Reynolds number for jet nozzle diameter and velocity equals 35 (air). The following boundary conditions were prescribed for velocity components u, v, w, kinetic energy of turbulence k and specific dissipation rate ω (in nondimensional form with 4.8 m/s and d as reference velocity and length): x = A (cross flow inlet): u = U e =.5, v = w = k = ω = k/(.1ν) z =, (x + y ).5 (jet nozzle): u = v =, w = 1. ( x + y /.5) 6 k = 3 1 ω = k/(.5ν) x = 4 (outlet): Neumann b.c. for u, v, w, k, ω z = (bottom wall): u = v = w = k = ω = 6ν/(β 1 y 1) 3

4 side and upper walls: inviscid (slip) wall, i.e. normal component of velocity =, Neumann b.c. for: tangential velocity, k, ω The lateral and upper walls were considered inviscid in order to reduce the number of grid points. This simplification is made possible by the fact, that the flow rate is prescribed not in terms of pressure drop in the channel but directly in terms of velocities, thus there is no need to capture friction on walls exactly, and further justified by the fact, that in the measurement measures to eliminate the displacement effect of neglected boundary layers were taken as well (slightly diverging walls). Figures 3 5 show profiles of horizontal and vertical velocity component in the mid-plane. The measured values [1] are for two inlet turbulence intensities and different runs. It is apparent that in the simulation the jet bents too slowly with respect to measurements. A test computation with slightly different heights of the domain showed no improvement. Thus we could say that this is not due to the inviscid forces but rather due to the inaccurate model for turbulent mixing. The model predicts higher vertical velocity than, which is caused by worse capturing of the secondary vortices the jet breaks in, see Fig. 6. Moreover, it seems according to measurements that the flow-field contains large scale unsteady vortices and the errors have more fundamental origin in the steady RANS approach. 5 Impinging jet flow 5.1 Isothermal case Here we consider the solution of RANS equations with and EARS turbulence models in the case of ERCOFTAC s impinging jet [1] for nozzle-to-wall distance nozzle diameters D and nozzle Reynolds number 3. In the previous works [5, 4] it has been found that eddy viscosity models highly over-predict the thickness of wall jet, not to mention the over-prediction of turbulence kinetic energy around impingement. The improvement brings the Bradshaw hypothesis used in the model as an eddy viscosity limiter. The model however has poor near wall behavior typical for k-ω models, apparent in Fig. 7, where the kink in wall shear stress at radial position r/d is missing. A k-ɛ model is capable of predicting the kink and when the -like eddy viscosity limiter is used, also the wall jet prediction is correct [4]. In the present work we consider the k-ω model only and show the influence of constitutive relation. Indeed, the improves the k-ω s behavior here. The and provide similar results in terms of radial velocity, Fig. 8, the Reynolds shear stress shown in Fig. 1 seems better in the model. The also attenuates the over-prediction of turbulence kinetic energy at impingement typical for eddy viscosity, see Fig. 9, however the measured value is still well lower. 5. Heat transfer In this section we consider the heat transfer from the heated impingement wall. Since the temperature difference of approx. 5 C as well as velocity is small, the energy equation is simplified to a passive scalar transport equation for temperature T : T t + u it = [ ] ν T u x i x i P r x i x it, (9) i where P r is laminar Prandtl number and u it turbulent heat transfer. Three closure models for u it were used. First one is the eddy diffusivity model u it = ν T P r T T x i, (1) where ν T is eddy viscosity and P r T turbulent Prandtl number set equal to.85. Then, with the anisotropy models for Reynolds stress we used two anisotropic models for heat transfer. The generalized gradient-diffusion hypothesis (GGDH) model reads u it = C t τu iu T j, C t =.3, (11) 4

5 z / D u / U j w / U j Fig. 3: Horizontal (left) and vertical (right) component of velocity, center-plane, x/d = 1 where τ is the turbulent time scale. In the fully turbulent region, τ = k/ɛ. The lower limit is given by Kolmogorov time scale ν/ɛ, whereas the realizability gives the upper limit. The final expression is [8] [ ( ) ] k ν τ = min max ɛ, 6 C R,, ɛ 6C µ II S C R =.46, C µ =.9. (1) The third model considered, which formally includes the previous two, is the one of Abe, Suga (AS) [9]. u it = c t τ(σ ij + α ij ) T, σ ij = c σ δ ij + c σ1 u iu j/k + + c σ u iu l u l u j/k, α ij = c α τω ij + + c α1 τ(ω il u l u j/k + Ω lj u iu l /k), (13) where the coefficients are.38 c t = (1 exp( R T /1)), 1/4 c σ =, c σ1 =.f b +.1f P r, c σ = 1 f b f P r, c α =, c α1 = (1 f P r ) [.5gA ξ 1 + 5ξ. ] exp[ (ξ/.) ], g A + (ξ +.) f b = (1 f P r ) exp[ (ξ/.) (g A /.3) ], f P r = (1 + (P r/.85) 1.5 ) 1, g A =.3(1 exp[ (R T /7) ]), ξ = τ S ij S ij, (14) and the turbulent time scale τ is taken same as in the, Eq. (4). 5.3 Heat transfer results The temperature of impingement wall is kept constant at T w = 313 K and the temperature of fluid at nozzle exit was T = 93 K. The Nusselt number is defined as Nu = D T T n w, (15) T w T 5

6 z / D u / U j w / U j Fig. 4: Horizontal (left) and vertical (right) component of velocity, center-plane, x/d = z / D u / U j w / U j Fig. 5: Horizontal (left) and vertical (right) component of velocity, center-plane, x/d = 6

7 Fig. 6: Secondary flow in the plane x = d, model (left) and (right) τ w / (ρ w b ) r / d Fig. 7: Wall shear stress on impingement wall where n denotes wall normal direction. The result in terms of Nusselt number is shown in Fig. 11. All models over-predict the heat transfer in the stagnation region. This allows us a statement that even a sofisticated constitutive relation for Reynolds stress as well as for turbulent heat transfer hardly guarantees quantitatively acceptable results. Instead, the turbulent time scale prediction is of higher importance. We recall here a very good results of Merci, Dick [7] achieved with eddy diffusivity heat flux model, but with a non-linear k-ɛ model, where attention has been payed both to the constitutive relation as well as to the ɛ-equation (i.e. time scale prediction). Thus we can conclude that a more complex constitutive relations remain here limited by turbulent scale prediction, here the ω-equation. As for constitutive relations, the eddy diffusivity assumption leads to even qualitatively incorrect result with no drop of heat transfer at r/d 1.5, analogous error as in wall friction. Other models predict the presence of the drop correctly, with GGDH even being in good agreement with experiment starting from this radial position. A simple mean to improve the time scale prediction here could be the well known Yap correction, which (originally in a k-ɛ model) increases 7

8 u / w b u / w b r/d =.5 r/d = u / w b u / w b r/d =.5 r/d = Fig. 8: Velocity in the impinging jet at different radial positions.5. k / w b Fig. 9: Turbulence kinetic energy k on impinging jet axis 8

9 .1.8 τ 13 / w b τ 13 / w b r/d =.5 r/d = τ 13 / w b..4.6 τ 13 / w b r/d =.5 r/d = Fig. 1: Reynolds shear stress in the impinging jet at different radial positions 35 3 ERCOFTAC + eddy diffusivity + GGDH + AS 5 Nu r / D Fig. 11: Nusselt number on the impingement wall 9

10 the source in the ɛ-equation proportionally to k and yw 3 and thus diminishing the near wall overprediction of k in the impingement region. We added the analogous term Y c in the ω-equation of (6) in the following form with the switching functions from [7] Dω Dt = γ ω k P C µ k Y cɛ, S 1.5Ω, Y cɛ =.13(1 f Ry ) k max (.4k 3/ 1, ), otherwise y ɛy S = S ij S ij, Ω = Ω ij Ω ij, f Ry = [ π sin ( { } ) ] Ry min max 5 3, 1, 1, ɛ = β kω, β C µ =.9. The influence of Yap correction in the and GGDH model on Nusselt number is shown in Fig. 1. While it is effective the improvement is insufficient. Same degree of change is observed also for the AS turbulent heat flux model. 6 Conclusions Several turbulence models were used to simulate turbulent momentum and heat transport in jet in cross-flow and in impinging jet flow. A good qualitative agreement with experimental data was achieved in both cases. While gives much better results than model for both flow cases, its accuracy remains limited by the turbulent time scale model which is very similar to that of model. This limitation is most apparent in the impinging jet heat transfer simulation. Acknowledgment This work was partially supported by grant No. 1/5/681 GA CR, and Research Plan MSM References [1] D. Cooper, D. C. Jackson, B. E. Launder, G. C. Liao, J. W. Baughn, X. Yan, and M. Masbah. Normally impinging jet from a circular nozzle. Classic ER- COFTAC Database at University of Surrey. [] Antti Hellsten. New advanced k-ω turbulence model for high-lift aerodynamics. In 4nd AIAA Aerospace Sciences Meeting. Reno, Nevada, 4. [3] K. Kozel, P. Louda, and J. Příhoda. Numerical solution of D and 3D incompressible viscous flow problems. In Internal Flows, 5th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows, Gdansk, 1. [4] K. Kozel, P. Louda, and J. Příhoda. Numerical solution of turbulent backward facing step and impinging jet flows. In T. Lajos and J. Vad, editors, Conference on modelling fluid flow, pages , Budapest, 3. [5] P. Louda. Numerické řešení dvourozměrného a třírozměrného turbulentního impaktního proudění. PhD thesis, FS ČVUT, Praha,. [6] F. R. Menter. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 3(8): , August [7] Bart Merci and Erik Dick. Heat transfer predictions with a cubic kɛ model for axisymmetric turbulent jets impinging onto a flat plate. Int. J. of Heat nad Mass Transfer, 46:469 48, 3. [8] Masoud Rokni and Thomas B. Gatski. Predicting turbulent convective heat transfer in fully developed duct flows. Int. J. of Heat and Fluid Flow, :381 39, 1. 1

11 35 3 ERCOFTAC + GGDH + GGDH + Yap cor. 5 Nu r / D Fig. 1: Influence of Yap correction on Nusselt number [9] K. Suga and K. Abe. Nonlinear eddy viscosity modelling for turbulence and heat transfer near wall and shear-free boundaries. Int. J. of Heat and Fluid Flow, 1:37 48,. [1] V. Uruba, O. Mazur, and P. Jonáš. The instabilities in jet-cross flow interaction. In International Conference Engineering Mechanics, volume IV, pages , Svratka,. [11] S. Wallin. Engineering turbulence modeling for CFD with a focus on explicit algebraic Reynolds stress models. PhD thesis, Royal Institute of Technology, Stockholm,. 11