This paper presents a gradient-based reconstruction approach for simulations of compressible single and multi-species Navier-Stokes equations. The novel feature of the proposed algorithm is the efficient reconstruction via derivative sharing between the inviscid and viscous schemes: highly accurate explicit and implicit gradients are used for the solution reconstruction expressed in terms of derivatives. The higher-order accurate gradients of the velocity components are reused to compute the viscous fluxes for efficiency and significantly improve the solution and gradient quality, as demonstrated by several viscous-flow test cases. The viscous schemes are fourth-order accurate and carefully designed with a high-frequency damping property, which has been identified as a critically important property for stable compressible-flow simulations with shock waves [Chamarthi et al., JCP, 2022]. Shocks and material discontinuities are captured using a monotonicity-preserving (MP) scheme, which is also improved by reusing the gradients. For inviscid test cases, The proposed scheme is fourth-order for linear and second-order accurate for non-linear problems. Several numerical results obtained for simulations of complex viscous flows are presented to demonstrate the accuracy and robustness of the proposed methodology.
We analyze the disturbed flow in the supersonic laminar boundary layer when local heated elements are placed on the surface. It is exhibited that these flows are described in terms of free interaction theory for specific sizes of thermal sources. We construct the numerical solution for flat supersonic problem in the viscous asymptotic layer in which the flow is described by nonlinear equations for vorticity, temperature with the interaction condition which provides influence of perturbations to the pressure in the main order.
In this paper we present analytical studies of three-dimensional viscous and inviscid simplified Bardina turbulence models with periodic boundary conditions. The global existence and uniqueness of weak solutions to the viscous model has already been established by Layton and Lewandowski. However, we prove here the global well-posedness of this model for weaker initial conditions. We also establish an upper bound to the dimension of its global attractor and identify this dimension with the number of degrees of freedom for this model. We show that the number of degrees of freedom of the long-tim…
The growth rate of the compressible Rayleigh-Taylor instability is studied in the presence of a background temperature gradient, $Θ$, using a normal mode analysis. The effect of $Θ$ variation is examined for three interface types corresponding to combinations of the viscous properties of the fluids (inviscid-inviscid, viscous-viscous and viscous-inviscid) at different Atwood numbers, $At$, and, when at least one of the fluids' viscosity is non-zero, as a function of the Grashof number. For the general case, the resulting ordinary differential equations are solved numerically; however, dispersi…
Flow instability and turbulent transition can be well explained using a new proposed theory--Energy gradient theory [1]. In this theory, the stability of a flow depends on the relative magnitude of energy gradient in streamwise direction and that in transverse direction, if there is no work input. In this note, it is shown based on the energy gradient theory that inviscid non-uniform flow is unstable if the energy in transverse direction is not constant. This new finding breaks the classical linear theory from Rayleigh that inviscid flow is unstable if the velocity profile has an inflection po…
Unsteady Lifting-Line Theory (ULLT) is a low order method capable of modeling interacting unsteady and finite wing effects at low computational cost. Most formulations of the method assume inviscid flow and small amplitudes. Whilst these assumptions might be suitable for small-amplitude aeroelastic problems at high Reynolds numbers, modern engineering applications increasingly involve lower Reynolds numbers, large amplitude kinematics and vortex structures that lead to aerodynamic non-linearities. This paper establishes that ULLT still provides a good solution for low Reynolds number, large-am…
We investigate quasi-two-dimensional relaxation, by surface tension, of a long straight stripe of inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. Combining analytical and numerical solutions, we describe the emergence of a self-similar dumbbell shape and find non-trivial dynamic exponents that characterize scaling behavior of the dumbbell dimensions.
In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations preserve initially imposed symmetry and that such symmetric flows are stable under general three-dimensional perturbations, globally in time. We work in three different contexts: two-and-a-half-dimensional, helical and axi-symmetric flows. In the inviscid case, we o…
Numerical simulations using the Lattice Boltzmann Method are presented of the following two-dimensional incompressible flow problem. Starting from configurations corresponding to translating inviscid equilibria, namely, the translating Föppl equilibria (counter-rotating point vortex pair, either fore or aft of a translating circular cylinder) and the translating Hill equilibria (counter-rotating point vortex pair, either fore or aft of an elliptic cylinder), viscosity is turned on for $t >0$ and the subsequent viscous interaction is simulated. The interaction is in a dynamically coupled settin…
The proper scale decomposition in flows with significant density variations is not as straightforward as in incompressible flows, with many possible ways to define a `length-scale.' A choice can be made according to the so-called \emph{inviscid criterion} \cite{Aluie13}. It is a kinematic requirement that a scale decomposition yield negligible viscous effects at large enough `length-scales.' It has been proved \cite{Aluie13} recently that a Favre decomposition satisfies the inviscid criterion, which is necessary to unravel inertial-range dynamics and the cascade. Here, we present numerical dem…
In this paper we investigate the asymptotic validity of boundary layer theory. For a flow induced by a periodic row of point-vortices, we compare Prandtl's solution to Navier-Stokes solutions at different $Re$ numbers. We show how Prandtl's solution develops a finite time separation singularity. On the other hand Navier-Stokes solution is characterized by the presence of two kinds of viscous-inviscid interactions between the boundary layer and the outer flow. These interactions can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover we apply the comp…
We carry out a systematic analytical and numerical study of spectral stability of discontinuous roll wave solutions of the inviscid Saint Venant equations, based on a periodic Evans-Lopatinski determinant analogous to the periodic Evans function of Gardner in the (smooth) viscous case, obtaining a complete spectral stability diagram useful in hydraulic engineering and related applications. In particular, we obtain an explicit low-frequency stability boundary, which, moreover, matches closely with its (numerically-determined) counterpart in the viscous case. This is seen to be related to but no…
We analyze both theoretically and experimentally the breakup of a pendant water droplet loaded with Sodium Dodecyl Sulfate (SDS). The free surface minimum radius measured in the experiments is compared with that obtained from a numerical solution of the full Navier-Stokes equations for different values of the shear and dilatational surface viscosities. This comparison shows the small but measurable effect of the surface viscous stresses on the system dynamics for sufficiently small spatiotemporal distances from the breakup point, and allows to establish upper bounds for the values of the shear…
Rayleigh showed that inviscid flow is unstable if the velocity profile has an inflection point in parallel flows. However, whether viscous flows is unstable or not is still not proved so far when there is an inflection point in the velocity profile. Fluid viscosity has showed dual role to the flow instability. In this paper, it is demonstrated for the first time that viscous parallel flow with inflectional velocity profile is sufficient for flow instability.
Solving compressible Navier-Stokes equations using the feature-enhanced neural network
School of Aeronautics, Northwestern Polytechnical University（西北工业大学航空学院）；International Joint Institute of Artificial Intelligence on Fluid Mechanics, Northwestern Polytechnical University（西北工业大学流体力学人工智能国际联合研究院）；National Key Laboratory of Aircraft Configuration Design（飞机构型设计全国重点实验室）
A special initial condition for (1+1)-dimensional Burgers equation is considered. It allows to obtain new analytical solutions for an arbitrary low viscosity as well as for the inviscid case. The viscous solution is written as a rational function provided the Reynolds number (a dimensionless value inversely proportional to the viscosity) is a multiple of two. The inviscid solution is expressed in radicals. Asymptotic expansion of the viscous solution at infinite Reynolds number is compared against the inviscid case. All solutions are finite, tend to zero at infinity and therefore are physicall…
In the physically non viscous fluid dynamics, "shock capturing" methods adopt either an artificial viscosity contribution or an appropriate Riemann solver algorithm. These techniques are necessary to solve the strictly hyperbolic Euler equations if flow discontinuities (the Riemann problem) must be solved. A necessary dissipation is normally used in such cases. An explicit artificial viscosity contribution is normally adopted to smooth out spurious heating and to treat transport phenomena. Such a treatment of inviscid flows is also widely adopted in the Smooth Particle Hydrodynamics (SPH) fini…
Shock reflection experiments are performed to study the large-scale convective mixing created by the forward jetting phenomenon. Experiments are performed at a wedge angle of $θ_{\mathrm{w}} = 30^{\circ}$ in nitrogen, propane-oxygen, and hexane with incident shock Mach numbers up to $M = 4$. Experiments are complimented by shock-resolved viscous simulations of triple point reflection in hexane for $M = 2.5$ to $6$. Inviscid simulations are performed over a wider range of parameters. Reynolds numbers up to $Re \lesssim 10^3$ are covered by simulations and Reynolds numbers of $Re \sim 10^5$ are …
Adopting the setting for the study of existence and scale locality of the energy cascade in 3D viscous flows in physical space recently introduced by the authors to 3D inviscid flows, it is shown that the anomalous dissipation is -- in the case of decaying turbulence -- indeed capable of triggering the cascade which then continues ad infinitum, confirming Onsager's predictions.
We present an asymptotic theory for analytical characterization of the high-Reynolds-number incompressible flow of a Newtonian fluid past a shear-free circular cylinder. The viscosity-induced modifications to this flow are localized and except in the neighborhood of the rear stagnation point, behave like a linear perturbation of the inviscid flow. Our theory gives a highly accurate description of these modifications by including the contribution from the most significant viscous term in a correctional perturbation expansion about an inviscid base state. We derive the boundary layer equation fo…
The new proposed "energy gradient theory," which physically explains the phenomena of flow instability and turbulent transition in shear flows and has been shown to be valid for parallel flows, is extended to curved flows in this study. Then, three important theorems for fluid dynamics are deduced. These theorems are (1) Potential flow (inviscid and irrotational) is stable. (2) Inviscid rotational (nonzero vorticity) flow is unstable. (3) Velocity profile with an inflectional point is unstable when there is no work input or output to the system, for both inviscid and viscous flows. These theor…
We study the stability of two-dimensional inviscid flows in an annulus between two porous cylinders with respect to three-dimensional perturbations. The basic flow is irrotational, and both radial and azimuthal components of the velocity are non-zero. The direction of the radial flow can be from the inner cylinder to the outer one (the diverging flow) or from the outer cylinder to the inner one (the converging flow). It had been shown earlier in Ref. \cite{IM2013a} that, independent of the direction of the radial flow, the basic flow can be unstable to small two-dimensional perturbations. In t…
A homogenization approach is proposed for the treatment of porous wall boundary conditions in the computation of compressible viscous flows. Like any other homogenization approach, it eliminates the need for pore-resolved fluid meshes and therefore enables practical flow simulations in computational fluid domains with porous wall boundaries. Unlike alternative approaches however, it does not require prescribing a mass flow rate and does not introduce in the computational model a heuristic discharge coefficient. Instead, it models the inviscid flux through a porous wall surrounded by the flow a…
The classical Prandtl-Batchelor theorem (Prandtl 1904; Batchelor 1956) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem to recirculating flows with quasi-periodic time dependence using ergodic and geometric analysis of Lagrangian dynamics. In particular, we show that 2D quasi-periodic viscous flows, in the limit of zero viscosity, cannot converge to recirculating inviscid flows with non-uniform vorticity distribution. A corollary of this result is that if the vorticity contou…
Providing evidence of finite-time singularities of the incompressible Euler equations in three space dimensions is still an unsolved problem. Likewise, the zeroth law of turbulence has not been proven to date by numerical experiments. We address this issue by high-resolution numerical simulations of the inviscid three-dimensional Taylor-Green vortex problem using a novel high-order discontinuous Galerkin discretization approach. Our main finding is that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but in…
We propose an invariant feature space for the detection of viscous dominated and turbulent regions (i.e., boundary layers and wakes). The developed methodology uses the principal invariants of the strain and rotational rate tensors as input to an unsupervised Machine Learning Gaussian mixture model. The selected feature space is independent of the coordinate frame used to generate the processed data, as it relies on the principal invariants of strain and rotational rate, which are Galilean invariants. This methodology allows us to identify two distinct flow regions: a viscous dominated, rotati…
The `Rayleigh line' $μ=η^2$, where $μ=Ω_o/Ω_i$ and $η=r_i/r_o$ are respectively the rotation and radius ratios between inner (subscript `i') and outer (subscript `o') cylinders, is regarded as marking the limit of centrifugal instability (CI) in unstratified inviscid Taylor-Couette flow, for both axisymmetric and non-axisymmetric modes. Non-axisymmetric stratorotational instability (SRI) is known to set in for anticyclonic rotation ratios beyond that line, i.e. $η^2<μ<1$ for axially stably-stratified Taylor-Couette flow, but the competition between CI and SRI in the range $μ<η^2$ has not yet b…
In the dynamics of viscous fluid, the case of vanishing kinematic viscosity is actually equivalent to the Reynolds number tending to infinity. Hence, in the limit of vanishing viscosity the fluid flow is essentially turbulent. On the other hand, the Euler equation, which is conventionally adopted for description of flow of inviscid fluid, does not possess proper turbulent behaviour. This raises the question of the existence of the passage to the limit of inviscid fluid for real low-viscosity fluids. To address this question, one should employ the theory of turbulent boundary layer near an infl…
We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic $\mathbb{T}^2$ setting and the case of a bounded channel $\mathbb{T}\times [0,1]$ with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity $ν\to 0$), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Ge…
We explore the behaviour of barotropic and irrotational fluids with a small viscosity under the effect of first-order acoustic perturbations. We discuss, following the extant literature, the difficulties in gleaning an acoustic geometry in the presence of viscosity. In order to obviate various technical encumbrances, when viscosity is present, for an extraction of a possible acoustic geometry, we adopted a method of double perturbations, whereby dynamical quantities such as the velocity field and potential undergo a perturbation both in viscosity and in an external acoustic stimulus. The resul…
In the non viscous fluid dynamics, Smooth Particle Hydrodynamics (SPH), as a free Lagrangian "shock capturing" method adopts either an artificial viscosity contribution or an appropriate Riemann solver technique. An explicit or an implicit dissipation, introduced in such techniques, is necessary to solve the Euler equations to solve flow discontinuities (the Riemann problem). Dissipation is useful to smooth out spurious heating and to treat transport phenomena. A simple, effective solution of the Riemann problem is here proposed, based on an empirical reformulation of the equation of state (Eo…
In this study, the kinetic inviscid flux (KIF) is improved and an implicit strategy is coupled. The recently proposed KIF is a kind of inviscid flux, whose microscopic mechanism makes it good at solving shock waves, with advantages against the shock instability phenomenon. When developing the implicit KIF, a phenomenon is noticed that the kinetic flux vector splitting (KFVS) part in boundary layers not only reduces the accuracy, but seriously reduces the Courant-Friedrichs-Lewy (CFL) number as well. As a result, in this paper, a new weight is proposed about how to combine the KFVS method well …
We present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier-Stokes system of equations. The system presented has a functional viscosity nu which depends on the pressure p=p(rho,mu_i) of the flow, with the density rho and the local concentration mu_i. High order Runge-Kutta time discretization techniques are employed, and different methods of dealing with arbitrary coupled boundary conditions are discussed. Analysis of the energy consistency of the scheme is performed in addition to inspection of the relative error of the solution compared to exact…
In physically inviscid fluid dynamics, "shock capturing" methods adopt either an artificial viscosity contribution or an appropriate Riemann solver algorithm. These techniques are necessary to solve the strictly hyperbolic Euler equations if flow discontinuities (the Riemann problem) are to be solved. A necessary dissipation is normally used in such cases. An explicit artificial viscosity contribution is normally adopted to smooth out spurious heating and to treat transport phenomena. Such a treatment of inviscid flows is also widely adopted in the Smooth Particle Hydrodynamics (SPH) finite vo…
We derive statistical equilibrium solutions of the truncated inviscid surface quasi-geostrophic (SQG) equations, and verify the validity of these solutions at late times in numerical simulations of the truncated SQG equations. The results indicate enstrophy thermalizes while energy can condense at the gravest modes, in agreement with previous indications of a direct cascade of enstrophy and an inverse cascade of energy in forced-dissipative SQG systems. At early times, the truncated inviscid SQG simulations show a behavior reminiscent of forced-dissipative SQG turbulence, and we identify spect…
The stability of a two-dimensional viscous flow between two rotating porous cylinders is studied. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the Reynolds number. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow…
