Numerical solution of the turbulent flow of an incompressible fluid in curved planar and axially symmetrical channels

The governing equations for axially symmetric flow, where the Reynolds stresses are expressed by scalar turbulent viscosity, are the Reynolds equations. The turbulence model k, ? is used in the well-known form for fully developed turbulent flow.The numerical method, a continuation of the MAC system¹, is adapted so that even for high Reynolds cell numbers precision (δx²) can be achieved for the steady flow. Irregular cells join the rectangular network on the curved surface. Von Neumann's stability condition of the linearised numerical system is investigated. Special problems concerning the numerical solution of the turbulence model equations are stated and a special procedure is worked out to ensure that the fields k, ? do not converge to physically meaningless values. The program for the computer is universal in that the boundary problems can be assigned by input data.As an example, an axially symmetrical diffuser with an area ratio of widening 1.40 is computed. Fields of velocity and pressure at the wall as well as fields v_T and k are assessed. The results are compared with an experiment. The conclusion is that this method is suitable for the problems mentioned in this study as well as for unsteady flow.     

Assessment of the validity of a log-law for wall-bounded turbulent bubbly flows

There has been considerable discussion in recent years concerning whether a log-law exists for wall-bounded, turbulent bubbly flows. Previous studies have argued for the existence of such a log-law, with a modified von Kármán constant, and this is used in various modelling studies. We provide a critique of this idea, and present several theoretical reasons why a log-law need not be expected in general for wall-bounded, turbulent bubbly flows. We then demonstrate using recent data from interface-resolving Direct Numerical Simulations that when the bubbles make a significant contribution to the channel flow dynamics, the mean flow profile of the fluid can deviate significantly from the log-law behaviour that approximately holds for the single-phase case. The departures are not surprising and the basic reason for them is simple, namely that for bubbly flows, the mean flow is affected by a number of additional dynamical parameters, such as the void fraction, that do not play a role for the single-phase case. As a result, the inner/outer asymptotic regimes that form the basis of the derivation of the log-law for single-phase flow do not exist in general for bubbly turbulent flows. Nevertheless, we do find that for some cases, the bubbles do not cause significant departures from the unladen log-law behaviour. Moreover, we show that if departures occur these cannot be understood simply in terms of the averaged void fraction, but that more subtle effects such as the bubble Reynolds number and the competition between the wall-induced turbulence and the bubble-induced turbulence must play a role.     

A control-volume based finite element method for three-dimensional incompressible turbulent fluid flow,heat transfer,and related phenomena

A control-volume based finite element method of equal-order type for three-dimensional incompressible turbulent fluid flow, heat transfer, and related phenomena is presented. The discretization equations are based mainly on the physics of the phenomena under consideration, more than on mathematical arguments. Special emphasis is devoted to the discretization of the convective terms and the continuity equation, and to the treatment of the boundary conditions imposed by the use of a high Reynolds k-?, type turbulence model. The pressure-velocity coupling in the fluid flow calculation is made from a derivative of the original SIMPLER method, without pressure correction. The discretized equations are solved in a sequential, rather than a coupled, form with significant advantage in the required computer time and storage. The method is an extension of a former version proposed by us for two-dimensional, laminar problems, and is here successfully applied to the following situations: three-dimensional deflected turbulent jet, and flows in 90° and 45° junctions of ducts with rectangular cross sections. The calculated results are in very good agreement with the experimental and numerical (obtained with the well established finite difference method) data available in the literature.     

Gas-particle two-phase turbulent flow in a vertical duct

Two-phase gas-phase turbulent flows at various loadings between the two vertical parallel plates are analyzed. A thermodynamically consistent turbulent two-phase flow model that accounts for the phase fluctuation energy transport and interaction is used. The governing equation of the gas-phase is upgraded to a two-equation low Reynolds number turbulence closure model that can be integrated directly to the wall. A no-slip boundary condition for the gas-phase and slip-boundary condition for the particulate phase are used. The computational model is first applied to dilute gas-particle turbulent flow between two parallel vertical walls. The predicted mean velocity and turbulence intensity profiles are compared with the experimental data of Tsuji et al. (1984) for vertical pipe flows, and good agreement is observed. Examples of additional flow properties such as the phasic fluctuation energy, phasic fluctuation energy production and dissipation, as well as interaction momentum and energy supply terms are also presented and discussed.

Applications to the relatively dense gas-particle turbulent flows in a vertical channel are also studied. The model predictions are compared with the experimental data of Miller & Gidaspow and reasonable agreement is observed. It is shown that flow behavior is strongly affected by the phasic fluctuation energy, and the momentum and energy transfer between the particulate and the fluid constituents.     

Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re = 800 stable?

A detailed case study is made of one particular solution of the 2D incompressible Navier–Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear-stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re = 800 is steady—and it is stable, to both small and large perturbations.     

Velocity gradients in a turbulent jet flow

We present a selection of results from experiments on an air turbulent jet flow, which included measurements of all the three velocity components and their nine gradients with the emphasis on the properties of invariant quantities related to velocity gradients (enstrophy, dissipation, enstrophy generation, etc.). This has been achieved by a 21 hot wire probe (5 arrays x 4 wires and a cold wire), appropriate calibration unit and a 3-D calibration procedure [1]. A more detailed account on the results will be published elsewhere.     

Numerical computation of turbulent flow through a square-sectioned 90° bend

Finite-volume, semi-elliptic computations are reported of the three-dimensional flow around a 90° square-sectioned bend for which detailed laser-Doppler measurements have been reported by Taylor et al.¹ While the standard k- eddy-viscosity model has been used in the main flow region, in place of the usual “wall functiond”, the mixing-length hypothesis has been employed to resolve the flow in the layer immediately adjacent to the wall. The scheme is successful in predicting the details of the primary and secondary flow fields both within the bend and downstream thereof.     

Segregated finite element algorithms for the numerical solution of large-scale incompressible flow problems

This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solution algorithms for the simulation of large-scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregated solution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solution algorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non-symmetric advection–diffusion type equations resulting from the segregated algorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solution algorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.     

A numerical study of turbulent flow over a two‐dimensional hill

Calculations of mean velocities and Reynolds stresses are reported for the recirculating flow established in the wake of two‐dimensional polynomial‐shaped obstacles that are symmetrical about a vertical axis and mounted in the water channel downstream of a fully developed channel flow for Re=6×10⁴. The study involves calculations of mean and fluctuating flow properties in the streamwise and spanwise directions and include comparisons with experimental data [Almeida GP, Durão DFG, Heitor MV. Wake flows behind two‐dimensional model hills. Experimental Thermal and Fluid Science 1993; 7: 87–101] for flow around a single obstacle with data resulting from the interaction of consecutive obstacles, using two versions of the low‐Reynolds number differential second‐moment (DSM) closure model. The results include analysis of the turbulent stresses in local flow co‐ordinates and reveal flow structure qualitatively similar to that found in other turbulent flows with a reattachment zone. It is found that the standard isotropization of production model (IPM), based on that proposed by Gibson and Launder [Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics 1978; 86(3): 191–511], with the incorporation of the wall reflection model of Craft and Launder [New wall‐reflection model applied to the turbulent impinging jet. AIAA Journal 1992; 32(12): 2970–2972] predicts the mean velocities quite well, but underestimates the size of the recirculation region and turbulent quantities in the shear layer. These inadequacies are circumvented by adopting a new cubic Reynolds stress closure scheme based on that more recently developed by Craft and Launder [A Reynolds stress closure designed for complex geometries. International Journal of Heat and Fluid Flow 1996; 17: 245–254] which satisfies the two component limit (TCL) of turbulence. In this model the geometry‐specific quantities, such as the wall‐normal vector or wall distance, are replaced by invariant dimensionless gradient indicators. Also, the model captures the diverse behaviour of the different components of the stress dissipation, ε_ij, near the wall and uses a novel decomposition for the fluctuating pressure terms. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical and experimental investigation of turbulent flow in a rectangular duct

Details of the turbulent flow in a 1:8 aspect ratio rectangular duct at a Reynolds number of approximately 5800 were investigated both numerically and experimentally. The three-dimensional mean velocity field and the normal stresses were measured at a position 50 hydraulic diameters downstream from the inlet using laser doppler velocimetry (LDV). Numerical simulations were carried out for the same flow case assuming fully developed conditions by imposing cyclic boundary conditions in the main flow direction. The numerical approach was based on the finite volume technique with a non-staggered grid arrangement and the SIMPLEC algorithm. Results have been obtained with a linear and a non-linear (Speziale) k–ε model, combined with the Lam–Bremhorst damping functions for low Reynolds numbers. The secondary flow patterns, as well as the magnitude of the main flow and overall parameters predicted by the non-linear k–ε model, show good agreement with the experimental results. However, the simulations provide less anisotropy in the normal stresses than the measurements. Also, the magnitudes of the secondary velocities close to the duct corners are underestimated. © 1998 John Wiley & Sons, Ltd.     

A typhoon-like vortex solution of incompressible 3D inviscid flow

This letter obtains a typhoon-like vortex solution by solving the Bragg-Hawthorne equation. The solution describes spiral paths of fluid material element on the Bernoulli surface, whereas some new exact solutions are obtained which are bounded in the whole region. The first one is a continued umbrella vortex solution, which is a typhoon-like vortex. The second one is a multi-planar solution, which is periodic in z-coordinate. Within each layer, there is an umbrella vortex solution similar to the first one. The above explicit solutions can be applied to the study of radial structure of typhoon. Both the solutions and the approaches used in the present work can also be applied to other complex flows.     

Verification of a mixed high‐order accurate DNS code for laminar turbulent transition by the method of manufactured solutions

This paper presents results on a verification test of a Direct Numerical Simulation code of mixed high‐order of accuracy using the method of manufactured solutions (MMS). This test is based on the formulation of an analytical solution for the Navier–Stokes equations modified by the addition of a source term. The present numerical code was aimed at simulating the temporal evolution of instability waves in a plane Poiseuille flow. The governing equations were solved in a vorticity–velocity formulation for a two‐dimensional incompressible flow. The code employed two different numerical schemes. One used mixed high‐order compact and non‐compact finite‐differences from fourth‐order to sixth‐order of accuracy. The other scheme used spectral methods instead of finite‐difference methods for the streamwise direction, which was periodic. In the present test, particular attention was paid to the boundary conditions of the physical problem of interest. Indeed, the verification procedure using MMS can be more demanding than the often used comparison with Linear Stability Theory. That is particularly because in the latter test no attention is paid to the nonlinear terms. For the present verification test, it was possible to manufacture an analytical solution that reproduced some aspects of an instability wave in a nonlinear stage. Although the results of the verification by MMS for this mixed‐order numerical scheme had to be interpreted with care, the test was very useful as it gave confidence that the code was free of programming errors. Copyright © 2009 John Wiley & Sons, Ltd.     

An implementation of the Spalart–Allmaras DES model in an implicit unstructured hybrid finite volume/element solver for incompressible turbulent flow

In this paper, we describe an implicit hybrid finite volume (FV)/element (FE) incompressible Navier–Stokes solver for turbulent flows based on the Spalart–Allmaras detached eddy simulation (SA‐DES). The hybrid FV/FE solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at vertices, making the current solver a staggered‐mesh scheme. The SA‐DES turbulence equation is solved after the velocity and the pressure fields have been updated at the end of each time step. The same matrix‐free FV method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity, which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA‐DES model in a hybrid flow solver. This paper will address important implementation issues for high‐Reynolds number flows where highly stretched elements are typically used. In addition, some aspects of implementing the SA‐DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flow past a flat plate and a high‐Reynolds number flow around a high angle‐of‐attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation. Copyright © 2008 John Wiley & Sons, Ltd.     

Interaction of a gas-droplet turbulent jet with a cocurrent high-velocity high-temperature gas flow

A mathematical model and a method for calculating a gas-droplet turbulent jet with allowance for velocity nonequilibrium and virtual mass of the condensed phase during turbulent fluctuations and also heat and mass transfer within the three-temperature scheme are developed. Methodical calculations are performed. The results of these calculations are in reasonable agreement with available experimental data. The structure of the gas-droplet jet in a cocurrent high-velocity high-temperature gas flow is studied by numerical methods. The ratio of intensities of heat and mass transfer between the phases and turbulent diffusion transfers of substances is found to be different at the initial, transitional, and basic segments of the jet. This difference is responsible for the nonmonotonic axial distribution of vapor density and the lines of the halved mass flow of the condensed phase. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 85–94, May–June, 2008.     

Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium

The boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium are considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the velocity parameter λ, which is the ratio of the external free stream velocity to the stretching surface velocity, permeability parameter of the porous medium k ₁, and Prandtl number Pr on the horizontal and transverse velocities, temperature profiles, surface heat flux and the wall shear stress, has been studied.     

High-order DG solvers for underresolved turbulent incompressible flows: A comparison of L2 and H(div) methods

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the underresolved regime, mass conservation and energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard L²-based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free H(div)-conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. This work raises the question whether and to which extent these two approaches are equivalent when applied to underresolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for underresolved simulations of turbulent flows due to their inherent dissipation mechanisms.     

Finite element solution of the streamfunction-vorticity equations for incompressible two-dimensional flows

The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework of the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution of two well-known benchmark problems: flow in a lid-driven cavity at Re ? 10,000 and flow over a backward-facing step at Re = 800.     

An analytical solution for steady flow of a Quemada fluid in a circular tube

An analytical solution is obtained for steady flow of Quemada-type fluids in a circular tube driven by a constant pressure gradient. Expressions are derived for velocity distribution and for volumetric flow rate as a function of pressure gradient or wall shear stress.     

二维定常不可压缩粘性流动N-S方程的数值流形方法

将流形方法应用于定常不可压缩粘性流动N-S方程的直接数值求解,建立基于Galerkin加权余量法的N-S方程数值流形格式,有限覆盖系统采用混合覆盖形式,即速度分量取1阶和压力取0阶多项式覆盖函数,非线性流形方程组采用直接线性化交替迭代方法和Nowton-Raphson迭代方法进行求解.将混合覆盖的四节点矩形流形单元用于阶梯流和方腔驱动流动的数值算例,以较少单元获得的数值解与经典数值解十分吻合.数值实验证明,流形方法是求解定常不可压缩粘性流动N-S方程有效的高精度数值方法.