Inviscid Problem Research Articles

Wave fields under the influence of a random-driven force: The Burgers equation

In this work, we examine the classical Burgers equation and investigate the effects of a random force on the wave field. Two scenarios are considered: the impact of a random force on different wave fields within the viscous Burgers equation and the effect of a periodic random force in the inviscid Burgers equation. For the first case, we demonstrate that the random force primarily causes wave fronts to increase or decrease depending on the dispersion parameter. For an initially deformed sinusoidal wave, the external force causes the mean wave field to spread out and dampen over time. The Cole-Hopf transformation is also used to obtain asymptotically the averaged wave field in certain regimes. For the inviscid problem, we assume the random force to be periodic with random phase to show that the mean wave field corresponds to the solution of the classical inviscid Burgers equation without external forces.

A New Fifth-Order Weighted Compact Nonlinear Scheme with Multi-Order Candidates Weighting for Hyperbolic Conservation Laws

ABSTRACT Based on one 5-point and two 2-point stencils, a nonlinear weighted scheme adaptive to the weighting of multi-order interpolation candidates for shock capturing is proposed. The JS weight function is adopted. The smoothness indicator of the 5-point stencil is specially designed with a higher-order leading term similar to the reference global smoothness indicator τ in the Z weighting. The design maintains that the nonlinear weights satisfy the sufficient condition for the scheme to avoid degradation at extreme points. The form of the weighting is also simplified, obviously reducing the weighting procedure's computational costs. Inviscid and viscous compressible flow problems are simulated to test the performance of the newly designed nonlinear weights based on the weighted compact nonlinear scheme. The numerical results show no obvious oscillations near the discontinuity, and the resolution of both discontinuities and smooth regions is better than JS, Z, and ZQ weighting.

Rayleigh–Taylor Flow with Two Interfaces: The Completed Boussinesq Approximation

A system of three horizontal fluid layers is considered, with two interfaces separating them. When the upper fluids are of higher density, the system is unstable and Rayleigh–Taylor instabilities occur, as interfacial disturbances grow with time and the fluids overturn. A linearized solution is presented for the corresponding inviscid problem. It reveals a neutrally stable situation when the fluid densities decrease with height. However, whenever a high density fluid lies above a less dense one, the linearized solution predicts exponential growth of the interface between them. With two interfaces present, several different flow scenarios are possible, depending on the two density ratios between the three fluids The interfacial waves can occur either in a sinuous or a varicose formation. A semi-numerical spectral method is used to obtain nonlinear solutions for three-layer viscous fluids, using a recently-published “Completed Boussinesq Approximation”. These nonlinear results are compared with the linearized inviscid solution and also with interface shapes obtained from an SPH algorithm. Results are shown for sinuous and varicose solution types, and inversion layer flows are discussed.

An improved hybridization strategy for the fifth-order unequal-sized weighted essentially non-oscillatory scheme

In this paper, a hybrid unequal-sized weighted essentially non-oscillatory (US-WENO) scheme is developed to reduce the computational cost caused by the computation of nonlinear weights and local characteristic decompositions in its procedures. First, an improved discontinuous sensor is designed based on the mathematical properties of the fourth-degree reconstruction polynomial in the US-WENO scheme. The proposed hybridization strategy can automatically and efficiently identify the troubled cells, and does not contain artificial parameters related to the problems. Second, the efficient linear scheme and the existing US-WENO scheme are combined to simulate the inviscid and viscous compressible flow problems. Finally, some benchmark numerical examples are used to verify the performance of the presented hybrid WENO scheme in various aspects, such as computational efficiency, low dissipation characteristics, shock capture ability, etc. The numerical experiments show that the proposed hybrid method can inherit all the features of the existing US-WENO scheme while improving its computational efficiency.

Improved perturbation solution for viscous flow in a dilating–contracting permeable channel with velocity slip

Recently, Mandal and Ghosh [“Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip,” Phys. Fluids 35, 047121 (2023)] constructed perturbation solutions for viscous flow in porous channels with a slip condition and moving walls restricted to slow wall dilation–contraction rates. Herein, we show that this “slowness” assumption may be completely removed. In doing so, we develop a more widely applicable and more accurate perturbation scheme for all dilation–contraction rates. Our strategy involves generating new and exact solutions to the linear, inviscid problem with slip condition, and then we draw on this precise form to construct more accurate perturbation expansions for solutions to the nonlinear flow model than are currently available.

Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface

Under consideration here are time-dependent three-dimensional stratified water flows of finite depth with a free surface and an interface (playing the role of an internal wave and separating two layers of constant and different densities). The main outcome of the paper is that, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, we prove that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. The rigid lid case is analyzed from the perspective of relaxed assumptions concerning the vorticity vectors. We also consider the viscous water wave problem case with normal and tangential stress conditions at the free surface and find that under similar conditions on the vorticity vectors, the conclusion about the two-dimensionality remains in place. Along the way, we also provide an explicit solution for the time-dependent three-dimensional inviscid water wave problem with an interface.

Stable, entropy-pressure compatible subsonic Riemann boundary condition for embedded DG compressible flow simulations

One approach to reducing the computational cost of simulating transitional compressible boundary layer flow is to adopt a near body reduced domain with boundary conditions enforced to be compatible with a computationally cheaper three-dimensional RANS simulation. In such an approach it is desirable to enforce a consistent pressure distribution which is not typically the case when using the standard Riemann inflow boundary conditions. We therefore revisit the Riemann problem adopted in many DG based high fidelity formulations.Through analysis of the one-dimensional linearized Euler equations it is demonstrated that maintaining entropy compatibility with the RANS simulation is important for a stable solution. It is also necessary to maintain the invariant for the Riemann outflow boundary condition in a subsonic flow leaving one condition that can be imposed at the inflow boundary. Therefore the entropy-pressure enforcement is the only stable boundary condition to enforce a known pressure distribution. We further demonstrate that all the entropy compatible inflow Riemann boundary conditions are stable providing the invariant compatible Riemann outflow boundary condition is also respected.Although the entropy-pressure compatible Riemann inflow boundary condition is stable from the one-dimensional analysis, two-dimensional tests highlight divergence in the inviscid problem and neutrally stable wiggles in the velocity fields in viscous simulations around the stagnation point. A two-dimensional analysis about a non-uniform baseflow assumption provides insight into this stability issue (ill-posedness) and motivates the use of a mix of inflow boundary conditions in this region of the flow.As a validation we apply the proposed boundary conditions to a reduced domain of a wing section normal to the leading-edge of the CRM-NLF model taken out of a full three-dimensional RANS simulation at Mach 0.86 and a Reynolds number of 8.5 million. The results show that the entropy-pressure compatible Riemann inflow boundary condition leads to a good agreement in pressure distribution.

Scalar auxiliary variable approache for the surface quasi-geostrophic equation

We present a numerical method for approximating the two-dimensional surface quasi-geostrophic equation. We first reformulate the equation into an equivalent system by using the scalar auxiliary variable approach. Then we propose first-order and second-order discretization schemes for solving the new surface quasi-geostrophic system in time. Furthermore, we prove that both of these schemes based on the scalar auxiliary variable approach satisfy an unconditionally energy stability property. Since the method gives two linear equations with constant coefficients in each time step. We have an efficient approach and accurate for solving these spacial fractional diffusion equations. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for inviscid problems and the problems with small viscosity.

On transonic boundary-layer receptivity over a vibrating flat plate coated with a thin liquid film

In this paper we study the generation of Tollmien–Schlichting waves initiated by vibrations of a wall where the wall is coated with a thin liquid film in a transonic flow regime. Motion of fluids are described by the two-dimensional Navier–Stokes equations assuming the Reynolds number is large. To find asymptotic solutions of the transonic boundary layer, we conduct an inspection analysis on the affine transformations of the triple-deck model for a subsonic flow and the unsteady full potential equations, with the intention of obtaining the order quantity of the free-stream Mach number in the transonic flow. We construct a modified triple-deck model for the transonic flow by considering the scalings of the perturbations that lead to the viscous–inviscid interaction problem for the flow in a subsonic regime. In particular, we are interested in the region where the subsonic scalings become invalid as the flow approaches transonic regime. We assume the wall oscillates in the vertical direction to the oncoming flow and these vibrations are periodic in time. We outline the process where the flow in the boundary layer converts the wall vibration perturbations into the instability modes which are measured by the receptivity coefficient. The viscous–inviscid interaction problem describes the stability of the boundary layer on the lower branch of the neutral curve. We show that the governing equations for the air viscous sublayer and the film flow are quasi-steady. The equation describing the inviscid layer of the airflow is unsteady and its referred to as the unsteady Kàrmàn–Guderley equation. The influence of the film surface tension is expressed through normal shear stress condition at the interface. We present an analytic formula for the amplitude of the Tollmien–Schlichting waves that are formed in the boundary layer. We analysed our model with different values of surface tension, initial film thickness and Kàrmàn–Guderley parameter. Depending on the value of these parameters, the initial amplitude of the instability waves may grow or decay.

A combined flux reconstruction and lattice Boltzmann flux solver for inviscid compressible flow simulation

AbstractA numerical method, which combines the high order flux reconstruction (FR) and the lattice Boltzmann flux solver (LBFS), has been developed to solve inviscid compressible flows in this article. Extensive research has shown that LBFS has absorbed the advantages of both finite volume method (FVM) and lattice Boltzmann method (LBM). On one hand, compared with FVM, FR can achieve arbitrary high order accuracy and is compact for parallel computing by avoiding wide stencils on meshes, which shows FR is a better choice than FVM. On the other hand, different from the discrete velocity Boltzmann equation based methods, the present method, that is, FR‐LBFS, solves the Euler equations essentially, and then the calculation efficiency can be improved. In FR‐LBFS, the common inviscid flux at the cell interface is evaluated by the local reconstruction of the lattice Boltzmann equation (LBE) solution from macroscopic flow variables at solution points. Unlike the traditional flux solution at the interface where the nonlinear convective terms have to be considered in the computation, the feature of linear convection term in LBE can make it easier to calculate flux in the current method. Furthermore, by considering real physical effects in LBM, the positive density and pressure can be achieved in the case of strong shocks and discontinuities. Some typical inviscid problems, including the advection of density perturbation, sod shock tube, Riemann problem for two‐dimensional gas dynamics, supersonic flow in a convergent nozzle with a ramp, subsonic flow around a cylinder, and transonic flow over a NACA0012 airfoil, are simulated to demonstrate the accuracies and robustness of the proposed FR‐LBFS.

A general Neural Particle Method for hydrodynamics modeling

Neural Particle Method (NPM) is a newly proposed Physics-Informed Neural Network (PINN) based, truly meshfree method for hydrodynamics modeling. In the NPM, PINN is applied to globally approximate field variables, and the high-order Implicit Runge–Kutta (IRK) method is used to treat the time integration. The NPM can easily achieve incompressibility and deal with the free-surface problem. However, the NPM is currently limited to inviscid hydrodynamics problems and is computationally expensive. In this work, we developed a general NPM (gNPM) for viscous hydrodynamics modeling. In the gNPM, a single pressure is output as the predicted pressure field rather than the multiple pressures in the original NPM. Thus, the size of the neural network is greatly reduced, making the gNPM computationally more efficient. Besides, the spatial derivatives in the governing equations are calculated with respect to the current spatial coordinates rather than the predicted space. In this manner, the gNPM is more straightforward to be implemented. Furthermore, by considering the viscous term in the conservation of momentum, the gNPM can be applied for viscous hydrodynamics modeling. The effectiveness and robustness of the gNPM have been demonstrated through several hydrodynamics benchmark cases with different boundary conditions. We highlight that the proposed gNPM is able to cope with highly uneven particle distributions, while the traditional meshfree method can produce severe failure.

A sliding-mesh interface method for three dimensional high order spectral difference solver

In this study a sliding-mesh interface method for three-dimensional unstructured mesh is developed and implemented to the high order spectral difference (SD) solver. The mortar approach is utilized for data exchange on the sliding interface for non-uniform unstructured mesh. The factors affecting the accuracy of the mortar method are analyzed, including mortar number, polynomial degree of mortar, non-uniformity of mortars, and mortar omission in polygon clipping. The method is validated with both inviscid and viscous problems with rotating flows. The numerical results show that the present method is able to retain the high order accuracy of the SD method. The influence of mortar omission in polygon clipping for poor quality complex mesh is discussed and evaluated with several validation problems. The errors with and without mortar omission are compared. The results show that the developed method is reliable and accurate even with mortar omission occurred occasionally in the simulation if the maximum omitted area is less than about 15% of its connected element on the sliding interface.

A two-stage fourth-order gas-kinetic CPR method for the Navier-Stokes equations on triangular meshes

An efficient gas-kinetic scheme with fourth-order accuracy in both space and time is developed for the Navier-Stokes equations on triangular meshes. The scheme combines an efficient correction procedure via reconstruction (CPR) framework with a robust gas-kinetic flux formula, which computes both the flux and its time-derivative. The availability of the flux and its time-derivative makes it straightforward to adopt an efficient two-stage temporal discretization to achieve fourth-order time accuracy. In addition, through the gas-kinetic evolution model, the inviscid and viscous fluxes are coupled and computed uniformly without any separate treatment for the viscous fluxes. As a result, the current scheme is more efficient than traditional explicit CPR methods with a separate treatment for viscous fluxes, and a fourth-order Runge-Kutta approach. Furthermore, a robust and accurate subcell finite volume limiting procedure is extended to the CPR framework for troubled cells, resulting in subcell resolution of flow discontinuities. Numerical tests demonstrate the high accuracy, efficiency and robustness of the current scheme in a wide range of inviscid and viscous flow problems from subsonic to supersonic speeds.

An efficient fifth-order finite difference multi-resolution WENO scheme for inviscid and viscous flow problems

In this paper, a more efficient fifth-order finite difference multi-resolution WENO scheme is designed to solve the compressible inviscid and viscous flow problems based on the original multi-resolution WENO scheme presented in Zhu and Shu (2018). This new improved method inherits all the following features of the original multi-resolution WENO scheme (Zhu and Shu , 2018): (1) a series of unequal-sized central stencils are used to obtain the spatial reconstruction polynomials; (2) any positive numbers whose sum is one can be set as the linear weights; (3) smaller L1 and L∞ errors could be obtained for smooth problems; (4) it is easier to extend to the unstructured finite volume framework. However, different from the original multi-resolution WENO scheme, this improved method simplifies the reconstruction process, significantly improves the computational efficiency, and has greater engineering application potential. Numerical results show that the two types of fifth-order finite difference multi-resolution WENO schemes have similar results, but the CPU time of this new multi-resolution WENO scheme is about 0.7 times that of the original one. Moreover, the new fifth-order multi-resolution WENO scheme with a small increase in the computational cost shows less dissipation error than the classical WENO scheme (Jiang and Shu, 1996), and can capture more subtle flow structures for solving inviscid and viscous flow problems on the same mesh level. Several benchmark inviscid and viscous problems are illustrated to verify the above conclusions and the improved performance of this fifth-order multi-resolution WENO scheme.

The generation of Tollmien\u2013Schlichting waves by free-stream turbulence in transonic flow

This paper studies the generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow over a half-infinite flat plate with a roughness element using an asymptotic approach. It is assumed that the Reynolds number (denoted Re) is large, and that the free-stream turbulence is uniform so it can be modelled as vorticity waves. Close to the plate, a Blasius boundary layer forms at a thickness of O(mathrm{{Re}}^{-{1}/{2}}), and a vorticity deformation layer is also present with thickness O(mathrm{{Re}}^{-{1}/{4}}). The report shows that there is no mechanism by which the vorticity waves can penetrate from the vorticity deformation layer into the classical boundary layer; therefore, a transitional layer is introduced between them in order to prevent a discontinuity in vorticity. The flow in the interaction region in the vicinity of the roughness element is then analysed using the triple-deck model for transonic flow. A novel asymptotic expansion is used to analyse the upper deck, which enables a viscous–inviscid interaction problem to be derived. In order to make analytical progress, the height of the roughness element is assumed to be small, and from this, we find an explicit formula for the receptivity coefficient of the Tollmien–Schlichting wave far downstream of the roughness.

A parameter-free total Lagrangian smooth particle hydrodynamics algorithm applied to problems with free surfaces

This paper presents a new Smooth Particle Hydrodynamics computational framework for the solution of inviscid free surface flow problems. The formulation is based on the Total Lagrangian description of a system of first-order conservation laws written in terms of the linear momentum and the Jacobian of the deformation. One of the aims of this paper is to explore the use of Total Lagrangian description in the case of large deformations but without topological changes. In this case, the evaluation of spatial integrals is carried out with respect to the initial undeformed configuration, yielding an extremely efficient formulation where the need for continuous particle neighbouring search is completely circumvented. To guarantee stability from the SPH discretisation point of view, consistently derived Riemann-based numerical dissipation is suitably introduced where global numerical entropy production is demonstrated via a novel technique in terms of the time rate of the Hamiltonian of the system. Since the kernel derivatives presented in this work are fixed in the reference configuration, the non-physical clumping mechanism is completely removed. To fulfil conservation of the global angular momentum, a posteriori (least-squares) projection procedure is introduced. Finally, a wide spectrum of dedicated prototype problems is thoroughly examined. Through these tests, the SPH methodology overcomes by construction a number of persistent numerical drawbacks (e.g. hour-glassing, pressure instability, global conservation and/or completeness issues) commonly found in SPH literature, without resorting to the use of any ad-hoc user-defined artificial stabilisation parameters. Crucially, the overall SPH algorithm yields equal second order of convergence for both velocities and pressure.

A novel stabilization method for high-order shock fitting with finite element methods

A moving-grid, shock-fitting, finite element method has been implemented that can achieve high-order accuracy for flow simulations with shocks. In this approach, element edges in the computational mesh are fitted to the shock front and moved with the shock throughout the simulation. The Euler or Navier-Stokes equations are solved on the moving mesh in an arbitrary Lagrangian-Eulerian framework. The method is implemented in two-dimensions in the context of a streamwise upwind Petrov-Galerkin finite element discretization with unstructured triangular meshes and mesh adaptation. It is shown that the shock interface motion equation has a wave nature, and disturbances can propagate along the shock interface. A SUPG stabilization term is introduced to the interface motion equation that is critical for ensuring that interface disturbances do not lead to non-convergent solution behavior. The formal order of accuracy of the scheme is verified, and the performance of the proposed scheme is assessed for both inviscid and viscous problems. It was found that the present scheme predicts smooth and noise-free surface heating for hypersonic flow over a cylinder with purely irregular triangular elements.

On Applying of Generalized Computational Experiment to Numerical Methods Verification

This work is devoted to the application of a generalized computational experiment for a comparative assessment of numerical methods accuracy. A generalized computational experiment allows one to obtain a numerical solution for a class of problems determined by the ranges of defining parameters variation. The approaches to the application of a generalized computational experiment in the presence of a reference solution and in its absence are dis-cussed. An example of constructing error surfaces is given when the solvers of the OpenFOAM software package are compared. The classic inviscid problem of oblique shock wave is used as a basic task. Variations of the key parameters of the problem — the Mach number and angle of attack — are considered. An example of the problem of flow around a cone at an angle of attack with varying Mach number, cone angle and angle of attack is also considered. The concept of an error index is introduced as an integral characteristic of deviations from the exact solution for each solver in the class of problems under consideration.

A note on the Stokes phenomenon in flow under an elastic sheet.

The Stokes phenomenon is a class of asymptotic behaviour that was first discovered by Stokes in his study of the Airy function. It has since been shown that the Stokes phenomenon plays a significant role in the behaviour of surface waves on flows past submerged obstacles. A detailed review of recent research in this area is presented, which outlines the role that the Stokes phenomenon plays in a wide range of free surface flow geometries. The problem of inviscid, irrotational, incompressible flow past a submerged step under a thin elastic sheet is then considered. It is shown that the method for computing this wave behaviour is extremely similar to previous work on computing the behaviour of capillary waves. Exponential asymptotics are used to show that free-surface waves appear on the surface of the flow, caused by singular fluid behaviour in the neighbourhood of the base and top of the step. The amplitude of these waves is computed and compared to numerical simulations, showing excellent agreements between the asymptotic theory and computational solutions. This article is part of the theme issue 'Stokes at 200 (part 2)'.

An efficient cell-centered finite-volume method with face-averaged nodal-gradients for triangular grids

In this paper, a face-averaged nodal-gradient approach is proposed as an efficient gradient method for a second-order cell-centered finite-volume discretization on triangular grids. The gradients needed in the linear reconstruction are computed in two steps: (1) compute gradients at nodes from solutions stored at cells, and (2) compute the gradient at each face by averaging the nodal gradients over the nodes forming the face. A unique feature of the proposed approach is that the same face-averaged gradient is used in linear reconstructions from both the left and right cells to the face centroid. This approach offers advantages over a conventional cell-gradient-based finite-volume method: reduced memory requirement for the gradients, smaller residual stencils, direct evaluations of viscous stresses and heat fluxes at solid boundaries, and less interpartition communication. The method is demonstrated for various inviscid and viscous problems from low to high Mach numbers on triangular grids.