Mean Flow Equations Research Articles

Quasi-Newton positivity-preserving scheme for the Spalart-Allmaras turbulence model using unstructured grids

A new implicit method for solving the Spalart-Allmaras (SA) turbulence model is proposed using a formal second-order upwind scheme on unstructured grids for steady-state flows. The mean-flow set of equations and the SA model equation are solved in a loosely coupled manner. While the Jacobian of the mean-flow equations is derived from a low-order approximation of the residual (i.e. the defect-correction approach), the SA Jacobian is obtained by modifying a direct linearization of the second-order residual. The modified Jacobian is designed to form an M-matrix without the artificial time derivative commonly used for steady-state problems. Namely, no time step is involved in solving the SA model equation. Thanks to a special design of the M-matrix, the positivity of the SA model working variable and the convergence of the SA model linearized problem (fixed-point iterative problem) is guaranteed. To further enhance the overall stability of the flow solver, the Runge-Kutta implicit algorithm is employed, allowing CFL numbers as high as one to two thousand for the mean-flow equations. A second-order upwind scheme is achieved with a least-square method using a limiter to suppress oscillation in the solution. However, second-order upwind scheme discretization of turbulence models may lack robustness, especially when using unstructured grids. The limiter can hinder the convergence of the non-linear iterations, especially of the turbulence model. A square root transformation is applied to the dependent variable of a baseline SA model to improve the non-linear convergence characteristics. Three variants of the SA model are implemented and tested together with two limiters. Numerical simulations of a broad range of aerodynamic applications are conducted. The transformed variant exhibits the least sensitivity to the limiter type.

A Novel Method for the Numerical Solution of a Hybrid Inverse Problem of Electrical Conductivity Imaging

A novel method for the numerical solution of a hybrid (coupled physics) inverse problem is proposed. Based on a regularized weighted mean curvature flow equation, this method can be considered as an alternative to the variational approach to solving weighted least gradient Dirichlet problems arising in electrical conductivity imaging, in particular, in Current Density Impedance Imaging (CDII). Utilizing the Sternberg–Ziemer theory, convergence of regularized solutions to a unique function of weighted least gradient is established. The numerical study is conducted to demonstrate the practicability and computational effectiveness of the proposed method.

On distributions of velocity random fields in turbulent flows

AbstractThe present paper aims to derive a partial differential equation (PDE) for the single‐time single‐point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non‐linear parabolic‐transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation (O. Reynolds, Philos. Trans. R. Soc. Lond. Ser. A 186 (1895), 123–164. http://doi.org/10.1098/rsta.1895.0004) and is a precise formulation of the PDF transport equation (S. B. Pope, Turbulent flows, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9780511840531.). The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed. Though the example is seemingly artificial, it demonstrates the PDF method based on the new PDF PDE.

Analysis of the kinetic energy recovery behind a tidal stream turbine for various submergence levels

Tidal turbines are commonly deployed at sea sites with water depths of up to 50 m to ease their deployment and quick maintenance operations. In these relatively shallow water depth conditions, the vertical expansion of tidal stream turbine wakes is restricted by the proximity of the rotor blades to the bottom bed and free-surface layer. These physical constrains can lead to changes in the flow mechanisms that drive momentum recovery behind the turbines, e.g. limiting the vertical fluxes of velocity. Understanding how the wake recovers depending on the submergence ratio is of utmost importance to designing the future multi-row tidal turbine arrays. Here, we adopt high-fidelity Large-Eddy Simulations (LES) with an Actuator Line Method (ALM) to represent the turbine's rotor to analyse the mean flow and transport equation for mean kinetic energy (MKE) behind a single bottom-fixed tidal turbine for four water depth values. Our results show that the close proximity of the turbine blade tip to the free-surface can notably constrain the wake expansion, with very shallow conditions leading to a limited contribution to the MKE replenishment of the turbulent momentum exchange over the vertical direction. Conversely, under such shallow conditions, the horizontal flux of MKE is enhanced over the lateral boundaries of the downstream wake. Our study evidences that the ratio of water depth to turbine diameter plays a relevant role in future tidal arrays and needs to be correctly accounted for in numerical models to provide reliable results.

Constraints on the wall layer in turbulent pipe flow

Dynamical constraints on the wall layer in turbulent pipe flow imply both a narrow peak in the streamwise component of the turbulent Lamb vector near the wall, and a scaling of the wall layer depth proportional to the depth of the viscous sublayer. An approximation of the Lamb vector distribution, which equates to the gradient of Reynolds stress, is proposed. Hence the equation for streamwise mean flow may be integrated to obtain an expression for the velocity profile in the wall layer.

On the transport equation for probability density functions of turbulent vorticity fields.

Vorticity random fields of turbulent flows (modelled over the vorticity equation with random initial data for example) are singled out as the main dynamic variables for the description of turbulence, and the evolution equation of the probability density function (PDF) of the vorticity field has been obtained. This PDF evolution equation is a mixed type partial differential equation (PDE) of second order which depends only on the conditional mean (which is a first-order statistics) of the underlying turbulent flow. This is in contrast with Reynolds mean flow equation which relies on a quadratic statistics. The PDF PDE may provide new closure schemes based on the first-order conditional statistics, and some of them will be described in the paper. We should mention that the PDF equation is interesting by its own and is worthy of study as a PDE of second order.

On the steady laminar boundary-layer flow over the family of rotating tori

We present the laminar boundary layer flow over the standard tori rotating in an otherwise quiescent incompressible fluid. The mean flow equations are derived by using a specific form of the toroidal–poloidal coordinate system. A pseudo-spectral collocation method mixed with the first-order Euler scheme is employed to solve the related steady mean flow equations. A fixed aspect ratio R is defined as the ratio of radii of the torus and its tube that distinguishes each fixed torus within the general family of standard tori. Tori of spindle type turn into a sphere when the aspect ratio R=0 and thus the mean flow results for the rotating sphere are consistently reproduced. The influence of aspect ratio R on the steady mean velocity components and the wall shear stresses are presented in detail. In a precise manner, the mean flow results for each rotating torus are explainable in a consistent way to the already established case of steady laminar boundary layer flow over the rotating sphere. The significance of the current work is discussed in terms of the hydrodynamic instabilities of boundary layer flows over the family of rotating tori.

Large-scale Structure and Turbulence Transport in the Inner Solar Wind: Comparison of Parker Solar Probe’s First Five Orbits with a Global 3D Reynolds-averaged MHD Model

Abstract Simulation results from a global magnetohydrodynamic model of the solar corona and solar wind are compared with Parker Solar Probe (PSP) observations during its first five orbits. The fully three-dimensional model is based on Reynolds-averaged mean-flow equations coupled with turbulence-transport equations. The model includes the effects of electron heat conduction, Coulomb collisions, turbulent Reynolds stresses, and heating of protons and electrons via a turbulent cascade. Turbulence-transport equations for average turbulence energy, cross helicity, and correlation length are solved concurrently with the mean-flow equations. Boundary conditions at the coronal base are specified using solar synoptic magnetograms. Plasma, magnetic field, and turbulence parameters are calculated along the PSP trajectory. Data from the first five orbits are aggregated to obtain trends as a function of heliocentric distance. Comparison of simulation results with PSP data shows good agreement, especially for mean-flow parameters. Synthetic distributions of magnetic fluctuations are generated, constrained by the local rms turbulence amplitude given by the model. Properties of this computed turbulence are compared with PSP observations.

Loosely coupled and coupled solution methods for the RANS equations and a one-equation turbulence model

The compressible Reynolds Averaged Navier-Stokes (RANS) equations represent a system of hyperbolic–parabolic nonlinear equations, which is composed of two different systems of equations. On the one hand, the mean flow equations represent the essential part, on the other hand, a system of equations is required to take the effects of turbulence into account. Variables and parameters of one system of equations are required to describe the other system of equations. In this sense, mean flow and turbulence flow equations are only one, fully coupled system of equations. On the other hand, the coupling between the systems of equations is often not as pronounced, which is why these systems are often weakly or loosely coupled solved in implementations. Against the background of these different perspectives, this article examines whether it is advantageous to solve the systems of equations in a mathematically consistent coupled or in a loosely coupled manner. A comparative study considering a broad range of applications is performed to discuss and demonstrate possible benefits when solving the sets of equations in either a fully coupled or a loosely coupled manner.

Unsteady continuous adjoint to URANS coupled with FW-H analogy for aeroacoustic shape optimization

In this paper, an unsteady aeroacoustic shape optimization tool is presented based on the continuous adjoint method. Aeroacoustic noise radiated by bodies in unsteady flows can be computed using a hybrid acoustic prediction tool, where the near-field flow and acoustics result from an unsteady CFD simulation while the acoustic propagation to far-field relies upon an acoustic analogy. The CFD solution is performed using the in-house GPU-enabled unsteady RANS solver for which a continuous adjoint solver is also available. Next to this, the noise prediction tool is developed based on the permeable version of the Ffowcs Williams and Hawkings analogy in the frequency domain and its implementation is verified by comparison to a well-known analytical solution of the sound field from a monopole source in uniform flow as well as comparing the results computed by the FW-H analogy with that of a time-accurate CFD run. The full mathematical formulation of the continuous adjoint method for the aforementioned acoustic analogy, coupled with the adjoint to the mean-flow and turbulence model equations, is presented. The accuracy of the gradients computed by the hybrid adjoint method is verified through comparisons with finite differences in two cases, both governed mainly by the tonal component in noise generation; these include the vortex shedding cylinder in a laminar flow and the rod-airfoil canonical benchmark in a turbulent flow. Finally, the programmed software is used to optimize the shape of the airfoil in the last test case, aiming at min. noise for a receiver located at far-field.

On the Mean Flow Solutions of Related Rotating Disk Flows of the BEK System

This paper investigates the effects of the YHP roughness model on the mean flow solutions of some flows belong to the family of the rotating BEK system flows. The governing mean flow equations are formulated in the rotating frame of reference, therefore, they include terms arising from the centrifugal force. These mean flow equations are solved using the method of lines and the backward difference method. Then, obtained results are compared for specifically selected value of roughness parameters with the results of a fundamentally different roughness model, the MW model. The results of the YHP model reveal that applying surface roughness changes the characteristics of the mean flow components. Moreover, the comparison of the YHP and MW models points that these changes are notably different for each model. Therefore, possible future researches can be conducted to investigate the stability characteristics of the flows due to the selection of the roughness model.

On the instability of Ekman boundary‐layer flow over a rough disk with a wavy surface profile

AbstractWe apply the method of lines and the backward difference method to solve highly nonlinear governing mean flow equations of the Ekman boundary‐layer flow over a rough disk with a wavy surface profile. These equations are derived using the YHP roughness model that imposes a wavy surface profile in terms of the radial position r, assuming rotational symmetry. Our main interest is to determine the effect of roughness on the local linear convective instability of the Ekman flow, and we solve the local linear stability equations of the disturbances stationary relative to the disk using the Chebyshev polynomials. The crossflow and viscous types of instability modes, the Type I and Type II modes respectively, are analyzed. Our theoretical findings prompts that an adequately designed rough disk can be a useful tool for a passive flow‐control method for the Ekman flow. The underlying physical mechanisms are investigated applying a fundamental energy analysis.

The formulation of the RANS equations for supersonic and hypersonic turbulent flows

AbstractAccurate prediction of supersonic and hypersonic turbulent flows is essential to the design of high-speed aerospace vehicles. Such flows are mainly predicted using the Reynolds-Averaged Navier–Stokes (RANS) approach in general, and in particular turbulence models using the effective viscosity approximation. Several terms involving the turbulent kinetic energy (k) appear explicitly in the RANS equations through the modelling of the Reynolds stresses in such approach, and similar terms appear in the mean total energy equation. Some of these terms are often ignored in low, or even supersonic, speed simulations with zero-equation models, as well as some one- or two-equation models. The omission of these terms may not be appropriate under hypersonic conditions. Nevertheless, this is a widespread practice, even for very high-speed turbulent flow simulations, because many software packages still make such approximations. To quantify the impact of ignoring these terms in the RANS equations, two linear two-equation models and one non-linear two-equation model are applied to the computation of five supersonic and hypersonic benchmark cases, one 2D zero-pressure gradient hypersonic flat plate case and four shock wave boundary layer interaction (SWBLI) cases. The surface friction coefficients and velocity profiles predicted with different combinations of the turbulent kinetic energy terms present in the transport equations show little sensitivity to the presence of these terms in the zero-pressure gradient case. In the SWBLI cases, however, comparisons show that inclusion of k in the mean flow equations can have a strong effect on the prediction of flow separation. Therefore, it is highly recommended to include all the turbulent kinetic energy terms in the mean flow equations when dealing with simulations of supersonic and hypersonic turbulent flows, especially for flows with SWBLIs. As a further consequence, since k may not be obtained explicitly in zero-equation, or certain one-equation, models, it is debatable whether these models are suitable for simulations of supersonic and hypersonic turbulent flows with SWBLIs.

Blending the Eddy-Viscosity and Reynolds-Stress Models Using Uniform High-Order Discretization

Accuracy, robustness, and efficiency are the three major goals of turbulence closure. Generally, eddy-viscosity models (EVMs) with the Boussinesq approximation are numerically stable and efficient. However, the accuracy of EVMs for aerodynamically complex configurations is inadequate. Reynolds-stress models (RSMs) are perceived as the most advanced turbulence models, but their robustness and efficiency could be improved. Here, a hybrid closure is proposed that combines a novel eddy-viscosity equation and six Reynolds-stress equations. The eddy-viscosity equation with natural boundary conditions on the walls is designed to implement a transition from eddy-viscosity mode (one-equation mode) to Reynolds-stress mode (seven-equation mode). The one-equation mode, where the eddy-viscosity equation is solved independently with the Boussinesq approximation and the Bradshaw relation, is used to establish the initial flowfield. After a given number of iterations, the seven-equation mode takes over and continues until convergence. The eddy-viscosity equation acts as a length-scale equation to close the RSM. The benchmark results show that the numerical stability is better than for traditional -based RSMs. Moreover, the stability allows a uniform high-order discretization of the mean-flow and turbulence equations (12 partial differential equations), which can reduce the cost needed to achieve a given level of accuracy. This improves the efficiency.

Effect of nozzle turbulent intensity in multiple round jets using openFOAM®

The present work is about the effects of turbulent intensity on the flow field of turbulent multiple jets. The simulations are carried out by using an open-source Computational Fluid Dynamics C++ code OpenFOAM®. The governing equations invoked are Reynolds Averaged Navier Stokes equations by using simpleFoam solver. The standard two-equation turbulence model is used along with the mean flow equations to account for turbulence effects. The solved flow field shows a significant effect of turbulent intensity on the potential core of both single and multiple jets which is supported by reference literature. Downstream the potential core, the turbulence intensity does not affect the decay rate of turbulent jets considered in the study. The performance parameters show that the higher initial values of turbulent intensity is favorable only for the single jet simulations. In contrast, the lower initial values of turbulent intensity is favorable for multiple jet simulations.

Turbulent bulk viscosity

The Reynolds decomposition is used to derive a new set of mean-flow equations for linear viscous fluids, with low compressibility (fluids in the liquid state). Additional shear and normal stresses are related to the velocity turbulent fluctuations in order to interpret the considerable energy dissipation in processes which are accompanied by rapid changes in liquid density, at least within the frame of the local thermodynamic equilibrium (LTE) approximation. A constitutive equation for the normal turbulent stresses, which involves the turbulent bulk viscosity, is given. A closure equation for the turbulent bulk viscosity is suggested. In the field of hydraulic engineering, the model is applied to investigate the classical problem of 1D finite amplitude pressure waves propagation in liquid-filled pipes (water hammer phenomenon). Available experimental pressure data are used to calibrate the model parameters and to demonstrate the capability of the proposed theoretical model in reproducing the experimental trials. The model results are discussed with attention paid to the role played by the turbulent bulk viscosity on the energy dissipation processes.

Reynolds analogy and turbulent heat transfer on rotating curved surfaces

To evaluate turbulent heat transfer on rotating turbine blades, knowledge of the Reynolds stress and heat conduction moments on the blades is required. This involves solving the mean flow equations of momentum and thermal energy on the rotating blades. In this paper, the turbine blades are represented by rotating curved surfaces, and turbulent heat transfer calculations are carried out by invoking a valid Reynolds analogy because of its simplicity. The classic Reynolds analogy is defined as the ratio between the eddy and thermal diffusivity in a stationary plane flow; however, its counterpart for rotating curved flows is still not known. To derive a Reynolds analogy for rotating curved flows, appropriate turbulence models to close the Reynolds transport equations for momentum and thermal energy are required. Assuming that, in the constant flux region, advection and diffusion of Reynolds stress and heat conduction moments are negligible compared to the production of turbulent energy, the Reynolds shear stresses and heat conduction moments can be solved in terms of the mean flow gradients, a turbulent Prandtl number (Prt)rc and a gradient Richardson number for rotating curved flows Rirc. These dimensionless numbers are dependent on mean flow properties, surface curvature, and system rotational speed. Thus derived, the classic Reynolds analogy can be recovered only when Rirc << 1 and (Prt)rc = 1. For turbine blade heat transfer, Rirc is not necessary very small and (Prt)rc differs from unity in the whole flow field. Therefore, using the classic Reynolds analogy to estimate turbine blade heat transfer will lead to incorrect result. The new Reynolds analogy, formulated for turbulent thermal flows on rotating curved surfaces, takes all pertinent external body force effects into account; thus, turbulent heat transfer evaluation on turbine blade surfaces can be substantially improved.

Integrability, exact reductions and special solutions of the KP–Whitham equations

Reductions of the KP–Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev–Petviashvili (KP) equation, are studied. Specifically, the soliton and harmonic wave limits of the KP–Whitham system are considered, which give rise in each case to a four-component (2+1)-dimensional hydrodynamic system. It is shown that a suitable change of dependent variables splits the resulting four-component systems into two parts: (i) a decoupled, independent two-component system comprised of the dispersionless KP equation, (ii) an auxiliary, two-component system coupled to the mean flow equations, which describes either the evolution of a linear wave or a soliton propagating on top of the mean flow. The integrability of both four-component systems is then demonstrated by applying the Haantjes tensor test as well as the method of hydrodynamic reductions. Various exact reductions of these systems are then presented that correspond to concrete physical scenarios.

An improved method for mean flow boundary conditions for computational aeroacoustics

An accelerated method of specifying Mean Flow Boundary Conditions for an unsteady flow is developed. Computation domains are inevitably finite in size, which means that its boundaries have to be located within the flow itself. This requires special boundary conditions which do not reflect outgoing waves while still being able to impose any incoming disturbances into the computation domain. As is well known, the Euler equations contain acoustic, vortical, and entropy waves, and all possible flows can be made by combining them. Because these waves are coupled, it is difficult to separate them into individual waves that the solver can modify. Moreover, it is desirable to set not just the waves but the mean of the combination of these waves at the boundaries and this takes a very long time. In this work, a new method for imposing the desired time-averaged mean flow at these computational boundaries is introduced. To reduce the complexity of the problem, but retaining the basic physics and difficulties, one-dimensional acoustic wave transmission is used to model wave propagation through a nearly choked nozzle. Mean and unsteady flow equations solved, simultaneously; and integrated with the existing non reflecting boundary conditions, and is able to quickly and accurately achieve the desired mean flow without spurious reflections of the outgoing disturbances, not only for deterministic, but chaotic flows.

Existence for a one-equation turbulent model with strong nonlinearities

The purpose of this article is to improve the existence theory for the steady problem of an one-equation turbulent model. For this study, we consider a very general model that encompasses distinct situations of turbulent flows described by the k-epsilon model. Although the boundary-value problem we consider here is motivated by the modelling of turbulent flows through porous media, the importance of our results goes beyond this application. In particular, our results are suited for any turbulent flows described by the k-epsilon model whose mean flow equation incorporates a feedback term, as the Coriolis force, the Lorentz force or the Darcy–Forchheimer’s drag force. The consideration of feedback forces in the mean flow equation will affect the equation for the turbulent kinetic energy (TKE) with a new term that is known as the production and represents the rate at which TKE is transferred from the mean flow to the turbulence. For the associated boundary-value problem, we prove the existence of weak solutions by assuming that the feedback force and the turbulent dissipation are strong nonlinearities, i.e. when no upper restrictions on the growth of these functions with respect to the mean velocity and to the turbulent kinetic energy, respectively, are required. This result improves, in particular, the existence theory for the classical turbulent k-epsilon model which corresponds to assume that both the feedback force and the production term are absent in our model.