Comparative Study of Algebraic Wall Model and Differential Equation Wall Model in Large Eddy Simulation of Turbulent Channel Flow

Turk. J. Math. Comput. Sci. 9(2018) 80–90 c MatDer http://dergipark.gov.tr/tjmcs http://tjmcs.matder.org.tr MATDER Comparative Study of Algebraic Wall Model and Differential Equation Wall Model in Large Eddy Simulation of Turbulent Channel Flow Muhammad Saiful Islam Mallika,∗ , Mohammad Azizul Hoquea,b , Md. Ashraf Uddinb a Department of Arts and Sciences, Ahsanullah University of Science and Technology, Dhaka-1208, Bangladesh. b Department of Mathematics, Shahjalal University of Science and Technology, Sylhet-3114, Bangladesh. Received: 02-08-2018 • Accepted: 06-12-2018 Abstract. A large eddy simulation (LES) is performed in a plane turbulent channel flow, where the near wall region is approximated by algebraic wall model (AWM) and differential equation wall model (DEWM). The simulation is performed by using a finite difference method of second order accuracy in space and a low-storage explicit Runge-Kutta method with third order accuracy in time. The computational results are compared with those from direct numerical simulation (DNS) data. Comparing the results throughout the calculation domain we have found that the results from the LES with DEWM (LES-DEWM) approach show closer agreement with the DNS results. 2010 AMS Classification: 76F65, 76D05. Keywords: Large eddy simulation, turbulent channel flow, algebraic wall model, differential equation wall model. 1. Introduction The study on turbulent channel flow [5, 7, 13–15, 18, 21, 25, 30] is promising not only for understanding turbulence phenomena but also for testing of validation of turbulence models in numerical simulation. There are varieties of turbulence simulation technique, such as DNS [7, 16, 18], LES [2–5, 7, 14, 15, 19, 24, 25, 29, 30], RANS (Reynoldsaveraged Navier-Stokes) and others. A simulation that resolves all the scales of turbulence motion is called DNS. But, due to its computational cost, still it is impractical for realistic engineering flows. On the other hand, RANS [6, 12] is the most used approximation to the solution of turbulent flow and this approach does not require large CPU resources. But one principal limitation of this approach is that the model used here must represent a very wide range of scales. An intermediate approach is the LES methodology. In LES, the computational cost is reduced by means of a filtering operation applied to the Navier-Stokes equations, thus eliminate many of the small scales below the filter width. This approach resolves explicitly the dynamics of the unsteady large scales of turbulence and models the smaller ones. Another approach in LES for the wall-bounded flows is to model the near-wall dynamics by wall stress models, e.g. Schumann model [21], Grötzbach model [7], Algebraic wall model [23], differential equation wall models [2–4, 27]. The main advantage of using such a model is that the resolution requirement for LES can be reduced significantly, thereby one can eliminate the computational cost much. Wall stress models supply wall shear stresses to the outer flow LES as a set of approximate boundary conditions. Another important issue for LES is the discretization method. A literature review suggests that for spatial discretization of the governing equations of LES the numerical methods *Corresponding Author Email addresses: (M.S.I. Mallik), (M.A. Hoque), (Md.A. Uddin) M.S.I. Mallik, M.A. Hoque, Md.A. Uddin , Turk. J. Math. Comput. Sci., 9(2018), 80–90 81 widely used are either spectral method or the conventional finite difference method with structured grids. Among these two methods, the conventional finite difference method [8, 17] is the most straightforward one. For time integration or for temporal discretization of the governing equations of LES the low-storage explicit Runge-Kutta methods are a popular choice. The low-storage Runge-Kutta methods [11, 28] call for minimum levels of memory locations during the time integration and can easily adapt with the modern large-scale scientific computing needs. Therefore, the objective of this study is to perform LES in a plane turbulent channel flow with near wall region approximation by AWM and DEWM. The spatial discretization of the governing equations of LES is done by a second order finite difference formulation, and for the temporal discretization a low-storage explicit Runge-Kutta method with third order accuracy is applied. For subgrid scale (SGS) modeling in LES the Standard Smagorinsky model (SSM) is used. Essential turbulence statistics based on these two LES approaches are calculated and compared with DNS data of Moser et al. [18]. Instantaneous streamwise shear velocity distribution at the immediate vicinity of the wall and instantaneous streamwise velocity distribution at the centerline plane of the channel are also computed from the two LES approaches and compared in different contour plots. Vortical structures in the computed flow field in 3D turbulent channel flow are visualized by iso-surfaces of second invariant of velocity gradient tensor. More specifically, the prime objective of this study is to compare the performance of AWM and DEWM in LES. 2. Governing Equations The governing equations of LES for an incompressible plane turbulent channel flow are the filtered Navier-Stokes and continuity equations. In Cartesian co-ordinates these equations can be written as ∂ 1 ∂ p̄ ∂  ∂ūi ∂u¯j
 ∂ūi + (ūi .u¯j + τi j ) = − + ν + ∂t ∂x j ρ ∂xi ∂x j ∂x j ∂xi ∂ūi =0 ∂xi where the index i, j = 1, 2, 3 refers to the x, y and z directions respectively. Associated to these directions, u¯x , u¯y and u¯z are the streamwise, wall normal and spanwise filtered velocity respectively. p̄ is the filtered pressure, ρ represents the fluid density and ν denotes the kinematic viscosity of the flow. τi j is SGS Reynolds stress tensor, which is unresolved term and must be modeled. The equations are non-dimensionalized by the channel half-width δ, and the wall shear velocity uτ . The Reynolds number of such a flow is therefore written as Reτ = uτ δ/ν . A schematic geometry of the plane channel flow and the co-ordinate system are shown in Figure 1. Figure 1. Schematic geometry of plane channel flow. In LES, the velocity field ui is decomposed into a filtered or large scale component ūi and a small or SGS component úi . This decomposition is done by applying a spatial filtering operation. Spatial filtering retains the large scale component to be resolved and remove the small scale component to be modeled. According to Sagaut [19], this decomposition is represented as: ui = ūi + úi . In this approach the effect of the SGS field appears through τi j , which is defined as τi j = ui u j − ūi u¯j . Comparative Study of Algebraic Wall Model and Differential Equation Wall Model in Large Eddy Simulation 82 Models used to approximate the τi j are called SGS models. Such a model represents the effect of the SGS field on the filtered filed. There are a number of SGS models. The most commonly used SGS mo (...truncated)