In a recent paper, Liu et al. [``Lift and drag in three-dimensional steady viscous and compressible flow'', Phys. Fluids 29, 116105 (2017)] obtained a universal theory for the aerodynamic force on a body in three-dimensional steady flow, effective from incompressible all the way to supersonic regimes. In this theory, the total aerodynamic force can be determined solely with the vorticity distribution on a single wake plane locating in the steady linear far field. Despite the vital importance of this result, its validity and performance in practice has not been investigated yet. In this paper, we performed Reynolds-averaged Navier-Stokes simulations of subsonic, transonic, and supersonic flows over a three-dimensional wing. The aerodynamic forces obtained from the universal force theory are compared with that from the standard wall-stress integrals. The agreement between these two formulas confirms for the first time the validity of the theory in three-dimensional steady viscous and compressible flow. The good performance of the universal formula is mainly due to the fact that the turbulent viscosity in the wake is much larger than the molecular viscosity therein, which can reduce significantly the distance of the steady linear far field from the body. To further confirm the correctness of the theory, comparisons are made for the flow structures on the wake plane obtained from the analytical results and numerical simulations. The underlying physics relevant to the universality of the theory is explained by identifying different sources of vorticity in the wake.
In this article, a suite of physically inconsistent properties of the Navier-Stokes equations, associated with the lack of mass diffusion and the definition of velocity, are presented. We show that these inconsistencies are consequences of the Lagrangian derivation that models viscous stresses rather than diffusion. A new model for compressible and diffusive (viscous and heat conducting) flows of an ideal gas, is derived in a purely Eulerian framework. We propose that these equations supersede the Navier-Stokes equations. A few numerical experiments demonstrate some differences and similaritie…
We present a quasi-two dimensional model of flowing soap films that bears striking similarity to the compressible Navier-Stokes equations. The variation in soap film thickness that is commonly used for flow visualization in experiments is analogous to density variations in the Navier-Stokes equations. When the soap film flow velocity is comparable to the Marangoni elastic wave velocity we recover the compressible Navier-Stokes equations and the soap film behaves like a two-dimensional isothermal viscous gas.
From the principle of least action the equation of motion for viscous compressible and charged fluid is derived. The viscosity effect is described by the 4-potential of the energy dissipation field, dissipation tensor and dissipation stress-energy tensor. In the weak field limit it is shown that the obtained equation is equivalent to the Navier-Stokes equation. The equation for the power of the kinetic energy loss is provided, the equation of motion is integrated, and the dependence of the velocity magnitude is determined. A complete set of equations is presented, which suffices to solve the p…
Multi-fractal scaling in the transition to the dissipative regime for fully-developed compressible turbulence is considered. The multi-fractal power law scaling behavior breaks down for very small length scales thanks to viscous effects. However, the effect of compressibility is found to extend the single-scaling multi-fractal regime further into the dissipative range. In the ultimate compressibility limit, thanks to the shock waves which are the appropriate dissipative structures, the single-scaling regime is found to extend indeed all the way into the full viscous regime. This result appears…
In a recent paper, Liu, Zhu and Wu (2015, {\it J. Fluid Mech.} {\bf 784}: 304) present a force theory for a body in a two-dimensional, viscous, compressible and steady flow. In this companion paper we do the same for three-dimensional flow. Using the fundamental solution of the linearized Navier-Stokes equations, we improve the force formula for incompressible flow originally derived by Goldstein in 1931 and summarized by Milne-Thomson in 1968, both being far from complete, to its perfect final form, which is further proved to be universally true from subsonic to supersonic flows. We call this…
Direct numerical simulation is performed to study compressible, viscous flow around a circular cylinder. The present study considers two-dimensional, shock-free continuum flow by varying the Reynolds number between 20 and 100 and the freestream Mach number between 0 and 0.5. The results indicate that compressibility effects elongate the near wake for cases above and below the critical Reynolds number for two-dimensional flow under shedding. The wake elongation becomes more pronounced as the Reynolds number approaches this critical value. Moreover, we determine the growth rate and frequency of …
Viscous and gravitational fingering refer to flow instabilities in porous media that are triggered by adverse mobility or density ratios, respectively. These instabilities have been studied extensively in the past for 1) single-phase flow (e.g., contaminant transport in groundwater, first-contact-miscible displacement of oil by gas in hydrocarbon production), and 2) multi-phase immiscible and incompressible flow (e.g., water-alternating-gas (WAG) injection in oil reservoirs). Fingering in multiphase compositional and compressible flow has received much less attention, perhaps due to its high c…
The compressible Neural Particle Method for Simulating Compressible Viscous Fluid Flows
Space and Astronautical Science, The Graduate University for Advanced Studies, SOKENDAI（空间与航天科学，高级研究大学，SOKENDAI）；Department of Aeronautics and Astronautics, Graduate School of Engineering, University of Tokyo（航空与航天系，工程研究生院，东京大学）
An immersed-boundary (IB) method is proposed and applied in the gas-kinetic BGK scheme to simulate incompressible/compressible viscous flow with stationary/moving boundary. In present method the ghost-cell technique is adopted to fulfill the boundary condition on the immersed boundary. A novel idea "local boundary determination" is put forward to identify the ghost cells, each of which may have several different ghost-cell constructions corresponding to different boundary segments, thus eliminating the singularity of the ghost cell. Furthermore, the so-called "fresh-cell" problem when implemen…
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…
Hydrodynamic interactions are transmitted by viscous diffusion and sound propagation: the temporal evolution of hydrodynamic interactions by both mechanisms is studied by direct numerical simulation in this paper. The hydrodynamic interactions for a system of two particles in a fluid are estimated by the velocity correlation of the particles. In an incompressible fluid, hydrodynamic interactions propagate instantaneously at the infinite speed of sound, followed by the temporal evolution of viscous diffusion. On the other hand, in a compressible fluid, sound propagates at a finite speed, which …
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…
In this paper, we investigate steady inviscid compressible flows with radial symmetry in an annulus. The major concerns are transonic flows with or without shocks. One of the main motivations is to elucidate the role played by the angular velocity in the structure of steady inviscid compressible flows. We give a complete classification of flow patterns in terms of boundary conditions at the inner and outer circle. Due to the nonzero angular velocity, many new flow patterns will appear. There exists accelerating or decelerating smooth transonic flows in an annulus satisfying one side boundary c…
We proposed rigorous definitions of Radon measure solutions for boundary value problems of steady compressible Euler equations which modeling hypersonic-limit inviscid flows passing two-dimensional ramps, and their interactions with still gas and pressureless jets. We proved the Newton-Busemann pressure law of drags of body in hypersonic flow, and constructed various physically interesting measure solutions with density containing Dirac measures supported on curves, also exhibited examples of blow up of certain measure solutions. This established a mathematical foundation for applications in e…
We study the evolution of a reactive field advected by a one-dimensional compressible velocity field and subject to an ignition-type nonlinearity. In the limit of small molecular diffusivity the problem can be described by a spatially discretized system, and this allows for an efficient numerical simulation. If the initial field profile is supported in a region of size l < lc one has quenching, i.e., flame extinction, where lc is a characteristic length-scale depending on the system parameters (reacting time, molecular diffusivity and velocity field). We derive an expression for lc in terms of…
We formulated a problem on hypersonic limit of two-dimensional steady non-isentropic compressible Euler flows passing a straight wedge. It turns out that Mach number of the upcoming uniform supersonic flow increases to infinite may be taken as the adiabatic exponent $γ$ of the polytropic gas decreases to $1$. We proposed a form of the Euler equations which is valid if the unknowns are measures and constructed a measure solution contains Dirac measures supported on the surface of the wedge. It is proved that as $γ\to1$, the sequence of solutions of the compressible Euler equations that containi…
