Time-dependent Reynolds-averaged Navier-Stokes Equations Research Articles

Three-dimensional numerical simulation of the operation of a rotating-detonation chamber with separate supply of fuel and oxidizer

A three-dimensional numerical simulation of the operation of an annular rotating-detonation chamber (RDC) with separate supply of combustible mixture components, hydrogen and air, is performed, and the calculation results are compared to available experimental data. The model is based on a system of time-dependent Reynolds-averaged Navier-Stokes equations complemented with a turbulence model and continuity and energy equations for a multicomponent reacting gas mixture. The system is solved using a coupled algorithm based on the finite volume method and particle method. Calculations are for the first time performed with allowance for effects of finite rates of turbulent and molecular mixing of the combustible mixture components with each other and with reaction and detonation products. The calculation results compare favorably with the experimental data obtained at the Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences.

Investigation of the Effects of Length to Depth Ratio on Open Supersonic Cavities Using CFD and Proper Orthogonal Decomposition

Simulations of supersonic turbulent flow over an open rectangular cavity are performed to observe the effects of length to depth ratio (L/D) of the cavity on the flow structure. Two-dimensional compressible time-dependent Reynolds-averaged Navier-Stokes equations with k-ω turbulence model are solved. A reduced order modeling approach, Proper Orthogonal Decomposition (POD) method, is used to further analyze the flow. Results are obtained for cavities with several L/D ratios at a Mach number of 1.5. Mostly, sound pressure levels (SPL) are used for comparison. After a reduced order modeling approach, the number of modes necessary to represent the systems is observed for each case. The necessary minimum number of modes to define the system increases as the flow becomes more complex with the increase in the L/D ratio. This study provides a basis for the control of flow over supersonic open cavities by providing a reduced order model for flow control, and it also gives an insight to cavity flow physics by comparing several simulation results with different length to depth ratios.

Numerical Modeling of Probable Maximum Flood Flowing through a System of Spillways

The results of a numerical model study of probable maximum flood (PMF) flow through a system of spillways consisting of an existing service spillway and a new auxiliary spillway are presented. A commercial computational fluid dynamics (CFD) code, Fluent, was used to solve the time-dependent Reynolds-averaged Navier-Stokes equations, a standard turbulence k-ε model with wall functions, and a water volume of fluid fraction equation. A two-dimensional approach velocity profile was used at the upstream inlet cross section. Water levels, flow splits between the existing and auxiliary spillways, and flow patterns were predicted and compared. A tentative design was chosen, constructed, and tested in a 1:54 scale physical model. Testing results were used to validate the CFD model. Results demonstrate that the CFD model is validated as accurate in the prediction of water levels in the reservoir, the integrated approach used is cost-effective and efficient in optimizing the designs of the auxiliary spillway, the tentative design cannot pass the PMF at the maximum pool level which suggests further modifications being necessary in the physical model.

Numerical Simulation on Stratified Flow over an Isolated Mountain Ridge

The characteristics of stratified flow over an isolated mountain ridge have been investigated numerically. The two-dimensional model equations, based on the time-dependent Reynolds averaged Navier-Stokes equations, are solved numerically using an implicit time integration in a fitted body grid arrangement to simulate stratified flow over an isolated ideally bell-shaped mountain. The simulation results are in good agreement with the existing corresponding analytical and approximate solutions. It is shown that for atmospheric conditions where non-hydrostatic effects become dominant, the model is able to reproduce typical flow features. The dispersion characteristics of gaseous pollutants in the stratified flow have also been studied. The dispersion patterns for two typical atmospheric conditions are compared. The results show that the presence of a gravity wave causes vertical stratification of the pollutant concentration and affects the diffusive characteristics of the pollutants.

Transport of Yellow Sand in Stratified Flow over an Isolated Mountain Ridge

The transport of yellow sand over mountain regions in the presence of internal gravity waves has been investigated numerically. The motion of yellow sand particles has been simulated in a Lagrangian frame of reference by solving the time-dependent Reynolds averaged Navier-Stokes equations. An implicit time integration in a fitted body grid arrangement was used to simulate the stratified flow over an isolated ideally bell-shaped mountain. The transport and deposition of particles of various sizes, and of the altitudes where particles were released have been analyzed. Particular attention was given to transport patterns of different sized particles in various atmospheric conditions. The results show that the particle size and the release altitude are both important factors in determining the trajectories of the particles. Small particles tend to be transported a long distance over the mountains, whereas heavier particles settle down around the release source. Due to the existence of an internal gravity wave, the particle release altitude affects the trajectory of the particles. The analysis and results provide a very useful tool for the study of atmospheric flow and transport of pollutants over real topographies.

Time accurate numerical study of turbulent supersonic jets

A time accurate numerical study is presented of an over-expanded Mach 2 circular turbulent jet in which the flow is assumed axisymmetric. The focus of this investigation is on the jet screech phenomenon resulting from the interaction between the large-scale turbulent mixing region instabilities and the regular spacing of the shock wave-expansion system, (shock cells), in the over-expanded jet. The solution is obtained of the “short” time-dependent Reynolds averaged Navier–Stokes equations (TRANS), using a two-equation, k– ω, turbulence model. The time accurate method was first calibrated for the given model geometry when the flow was fully expanded, and the resulting mean flow characteristics were compared with experimental data. The results were in broad agreement for the first 10 diameters of the jet downstream of the exit. Further downstream the time-averaged axial velocity decayed at a slightly faster rate than in the experiments. In an ideal inviscid fully expanded jet no shock cells would be present but in the turbulent jet calculations weak shock cells appeared which gradually died out beyond about 10 diameters from the nozzle exit. The calculated non-dimensional time-averaged transverse velocity profiles showed self-similarity, when allowance was made for the false origin of the shear layer, in agreement with the measured results. In the calculations for the over-expanded jet it was found, in agreement with experimental data, that the interaction between shock cell modulated instability waves and the shock-expansion system generated jet screech. It was found, as part of the screech phenomenon, that the shocks and the shock cells oscillated over a small distance which increased from the axis to a maximum within the shear layer. This shock unsteadiness resulted in the shocks being smeared when viewed in the equivalent steady flow calculations.

Numerical predictions of a rearward-facing-step flow in a supersoniccombustor

A computational fluid dynamics (CFD) code has been developed to compute the mixing and combustion of hydrogen fuel in the turbulent flowfields of supersonic combustion ramjets (scramjet) with primary interest for the case of transversely injected fuel. The code solves the three-dimensional time-dependent Reynolds-averaged Navier-Stokes equations, including species transport for a four-species, two-reaction, global finite-rate-chemistry model. Numerical integration was obtained by MacCormack's explicit method, and turbulence was modeled by the eddy viscosity model of Baldwin-Lomax. Mass diffusion was based on Pick's Law, and unit Lewis number was assumed. The code has been validated by comparing computational results with existing experimental data and applied to simulate a realistic scramjet combustor flowfield, particularly in the vicinity of the fuel injectors. Results from the comparisons are presented and discussed.

COMPUTATION OF INCOMPRESSIBLE TURBULENT FLOWS BY AN OPPOSED-DIFFERENCING SCHEME

A calculation scheme to predict incompressible turbulent flows in arbitrary shapes is presented. The procedure is based on the solution of the primitive-variable formulation of the time-dependent Reynolds-averaged Navier-Stokes equations in general curvilinear coordinates. An ordinary computational cell is used for continuity and transport balances, and all the physical properties are stored at the center of this element. The scheme uses an overlapping mesh along with forward and backward differencing for mass and pressure gradients, respectively. In a manner different from the staggered-grid approach, this procedure prevents the oscillatory behavior of the pressure field. The k-ϵ model is used to describe the turbulent flow process with particular attention to the boundary conditions. Computed results are compared with numerical and experimental data.

Computation of Separated Transonic Turbulent Flows

The two-dimensional Reynolds averaged compressible Navier-Stokes equations are solved using MacCormack's second-order accurate explicit finite difference method to simulate the separated transonic tur- bulent flowfield over an airfoil. Four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flowfield. each of these studies the time-dependent Reynolds averaged Navier-Stokes equations for two-dimensional compressive flow are used and tur- bulence closure is achieved by means of model equations for the Reynolds stresses. Wilcox1'2 used a first-order accurate numerical scheme and the two equation differential tur- bulence model of Saffman 12 to simulate the supersonic shock boundary-layer interaction experiment of Reda and Mur- phy 13 and the compression corner flow of Law.14 Good quan- titative agreement with the Reda and Murphy data was ob- tained, but only the qualitative features of the compression corner flow were well simulated. Using a more sophisticated second-order accurate numerical scheme, Baldwin3'4 con- sidered both the two equation differential model of Saffman and a simpler algebraic mixing-length model to simulate the hypersonic shock boundary-layer interaction experiment of Holden.15 He found the more elaborate model of Saffman to yield somewhat better results than the algebraic model, but at the cost of considerably more computing time. Good quan- titative agreement with experiment was not obtained with either model. Following Baldwin's approach all subsequent investigations have been performed using the more rigorous second-order accurate numerical scheme of Mac- Cormack.17'18 Deiwert5'6'11 considered an algebraic mixing- length model to simulate the transonic airfoil experiment of McDevitt et al. 16 while Horstman et al. 8 used a similar ap- proach to simulate their hypersonic shock boundary-layer ex- periment on an axisymmetric cylinder. In each of these studies, while qualitative features of the flows were described well, good quantitative agreement with experiment in the in- teraction regions was not obtained. Using a relaxing turbulence model Shang and Hankey7 simulated the compression corner flow of Law, and Baldwin and Rose10 simulated the flat plate flow of Reda and Murphy. In each of these studies the relaxing model was found to per- form significantly better than the simpler algebraic model and, according to Shang and Hankey, provided significantly better comparisons with measurements than were obtained by Wilcox using the two equation differential model of Saffman. In each of these studies it was essential that the full Navier- Stokes equations be considered to describe the viscous- inviscid interaction and the elliptic nature of separating-