On the smallest scale for the incompressible Navier-Stokes equations

We prove that, for solutions to the two- and three-dimensional incompressible Navier-Stokes equations, the minimum scale is inversely proportional to the square root of the Reynolds number based on the kinematic viscosity and the maximum of the velocity gradients. The bounds on the velocity gradients can be obtained for two-dimensional flows, but have to be assumed in three dimensions. Numerical results in two dimensions are given which illustrate and substantiate the features of the proof. Implications of the minimum scale result, to the decay rate of the energy spectrum are discussed.Research was supported in part by the National Acronautics and Space Administration under NASA Contract No. NAS1-18107, while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. Additional support was provided by the National Science Foundation under Grant DMS-8312264 and the Office of Naval Research under Contract N-00014-83-K-0422.     

Simulation of leading-edge vortex flows

An implicit upwind-relaxation finite-difference algorithm solving the incompressible Navier-Stokes equations is employed to simulate low-speed, three-dimensional, laminar, leadingedge vortex flows over three round-edged low-aspect-ratio wings. The effects of grid density, angle of attack, Reynolds number, and wing planform on the flowfield structures and integral values are studied. Computed results are presented and compared with experimental data.The work of the first author was supported by the NASA Langley Research Center under Contract NAS1-18585.     

Entropy jump across an inviscid shock wave

The Shock jump conditions for the Euler equations in their primitive form are derived by using generalized functions. The shock profiles for specific volume, speed, and pressure and shown to be the same, however, density has a different shock profile. Careful study of the equations that govern the entropy shows that the inviscid entropy profile has a local maximum within the shock layer. We demonstrate that because of this phenomenon, the entropy propagation equation cannot be used as a conservation law.This research was supported in part under NASA Contract No. NAS1-19480, while the second author was in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681-0001, U.S.A.     

The structure and evolution of the vorticity and temperature fields in thermals

The vorticity and temperature fields in axisymmetric thermals are studied via a projection method for the Navier-Stokes Boussinesq equations. Development of small-scale structure is observed at high Grashof numbers, and spreading rate, entrainment, mixing in the core, and generation of countersign vorticity are enhanced.This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48. Partial support was provided by the Applied Mathematical Sciences Program of the Office of Energy Research under Contract No. W-7405-Eng-48 and by the Defense Nuclear Agency under IACRO 90-824.     

Structure and stability of a laminar diffusion flame in a compressible,three-dimensional mixing layer

In this paper we consider the structure and stability of a three-dimensional, finite rate, reacting compressible mixing layer lying between two streams of reactants with different freestream speeds and temperatures. Using both numerical calculations and asymptotic analyses, the structure of the ignition and diffusion flame regimes was studied; in particular the effect of crossflow on the structure. The results of both approaches are in good agreement. In the stability analysis the general case of three-dimensional disturbances was studied. It was shown that certain general results, in particular the circle theorem, could be easily extended to three-dimensional disturbances in a compressible fluid with crossflow. The introduction of a chemical reaction, in the form of a flame sheet, was found to have complex effects on flow stability. These included the appearance of additional neutral modes compared with the nonreacting case and substantial changes in growth rates with heat release, the skew angle of the mean flow, and the direction of propagation of the disturbance wave. These results are discussed in detail.The first author was supported, in part, by AFOSR Contract 91-0180, and in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-19480 while in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681-0001, U.S.A. The second author was supported, in part, by AFOSR Contract 91-0250, and in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-19480 while in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681-0001, U.S.A.     

A model for adiabatic supersonic turbulent boundary layers

An asymptotic analysis of the equations describing supersonic turbulent flow over an adiabatic wall is carried out for high Reynolds numbers, Re, and mainstream Mach numbers, M _e=O(1). A general expression for the adiabatic-wall temperature is derived. The asymptotic theory constrains the types of turbulence models that are suitable to represent the effects of viscous dissipation. A simple algebraic turbulence model is proposed and comparisons with measured total enthalpy profile data show good agreement, capturing the overshoot observed in total enthalpy near the boundarylayer edge.This work was supported by NASA Langley Research Center under Grant NAG-1-832 and the Air Force Office of Scientific Research under Grants AFOSR-91-0069 and F49620-93-0130; Dr. Ruban was supported by a grant from United Technologies Corporation.     

Direct simulation of compressible turbulence in a shear flow

The purpose of this study is to investigate compressibility effects on the turbulence in homogeneous shear flow. We find that the growth of the turbulent kinetic energy decreases with increasing Mach number—a phenomenon which is similar to the reduction of turbulent velocity intensities observed in experiments on supersonic free shear layers. An examination of the turbulent energy budget shows that both the compressible dissipation and the pressure-dilatation contribute to the decrease in the growth of kinetic energy. The pressure-dilatation is predominantly negative in homogeneous shear flow, in contrast to its predominantly positive behavior in isotropic turbulence. The different signs of the pressure-dilatation are explained by theoretical consideration of the equations for the pressure variance and density variance. We previously obtained the following results for isotropic turbulence: first, the normalized compressible dissipation is of O(M _t ² ), and, second, there is approximate equipartition between the kinetic and potential energies associated with the fluctuating compressible mode. Both these results have now been substantiated in the case of homogeneous shear. The dilatation field is significantly more skewed and intermittent than the vorticity field. Strong compressions seem to be more likely than strong expansions.Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665, U.S.A.     

High-order accurate and high-resolution upwind finite volume scheme for solving euler/reynolds-averaged Navier-Stokes equations

A high-order accurate explicit scheme is proposed for solving Euler/Reynolds-averaged Navier-Stokes equations for steady and unsteady flows, respectively. Baldwin-Lomax turbulence model is utilized to obtain the turbulent viscosity. For the explicit scheme, the Runge-Kutta time-stepping methods of third orders are used in time integration, and space discretization for the right-hand side (RHS) terms of semi-discrete equations is performed by third-order ENN scheme for inviscid terms and fourth-order compact difference for viscous terms. Numerical experiments suggest that the present scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to numerical solution, even to unsteady problem. The project supported by the National Natural Science Foundation of China under Contract No. 59576007 and 19572038     

INTEGRATION METHOD FOR THE DYNAMICS EQUATION OF RELATIVE MOTION OF VARIABLE MASS NONLINEAR NONHOLONOMIC SYSTEM

1.IntroductionMoreandmoreattentionhasbeenpaidtothestudyofdynamicsofcomplicatedsystemwiththedevelopmentofmodernscienceandtechnology.Thestudyoftherelativemotionofvariablemasssystembyusingthetheoryandmethodofanalyticalmechanicsnotonlycanunifytheexpressionformbutalsocandisplayitssuperioritytothecomplicatedsystem.In1961,thedynamicsofrelativemotionofholonomicsystemwasderivedbyLur'ell].Inrecehtyears,LiulZIandLuol3'4]havegiventhedynamicsequationsofrelativemotionofvariablemassnonholonomicsystem.Howe…     

Linear instability of the supersonic wake behind a flat plate aligned with a uniform stream

In this paper we present a theoretical and numerical study of the growth of linear disturbances in the high Reynolds number laminar compressible wake behind a flat plate which is aligned with a uniform stream. No ad hoc assumptions are made as to the nature of the undisturbed flow (in contrast to previous investigations) but instead the theory is developed rationally by use of proper wake profiles which satisfy the steady equations of motion. The initial growth of near-wake perturbations is governed by the compressible Rayleigh equation which is studied analytically for long and short waves. These solutions emphasize the asymptotic structures involved and provide a rational basis for a nonlinear development. The phenomenon of enhanced stability with increasing Mach number observed in compressible free shear-layers is demonstrated analytically for short- and long-wavelength disturbances. The evolution of arbitrary wavelength perturbations is addressed numerically and spatial stability solutions are presented that account for the relative importance of the different physical mechanisms present, such as three-dimensionality, increasing Mach numbers, and the nonparallel nature of the mean flow. Our findings indicate that for low enough (subsonic) Mach numbers, there exists a region of absolute instability very close to the trailing edge with the majority of the wake being convectively unstable. At higher Mach numbers (but still not large—hypersonic) the absolute instability region seems to disappear and the maximum available growth rates decrease considerably. Three-dimensional perturbations provide the highest spatial growth rates.This work was carried out while the author was a summer visitor at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center under NASA Contract No. NAS1-18605.     

The principles of least action of variable mass nonholonomic nonconservative system in noninertial reference frames

This paper presents the generalized principles of least action of variable massnonholonomic nonconservative system in noninertial reference frame, proves theequivalence between Holder form and Suslov form, and then obtains differential equationsof motion of variable mass nonholonomic nonconservative system in noninertial referenceframe.     

ROUTH’S EQUATIONS FOR GENERAL NONHOLONOMIC MECHANICAL SYSTEMS OF VARIABLE MASS

In this paper,Routh’s equations for the mechanical systems of the variable masswith nonlinear nonholonomic constraints of arbitrary orders in a noninertial referencesystem have been deduced not from any variational principles,but from the dynamicalequations of Newtonian mechanics.And then again the other forms of equations fornonholonomic systems of variable mass are obtained from Routh’s equations.     

Smallest scale estimates for the Navier-Stokes equations for incompressible fluids

We consider solutions of the Navier-Stokes equations for incompressible fluids in two and three space dimensions. We obtain improved estimates, in the limit of vanishing viscosity, for the Fourier coefficients. The coefficients decay exponentially fast for wave numbers larger than the square root of the maximum of the velocity gradients divided by the square root of the viscosity. This defines the minimum scale, the size of the smallest feature in the flow.The work of Kreiss was supported in part by National Science Foundation under Grant DMS-8312264 and Office of Naval Research under Contract N-00014-83-K-0422.     

Computation of free surface waves around an arbitrary body by a Navier-Stokes solver using the psuedocompressibility technique

A Navier-Stokes equation solver is developed for computing free surface wave and viscous flow around an arbitrary body, in which a free surface model is introduced into the pseudocompressibility solution. The governing equations are classified in a vectorial form, with primitive variables, and a block diagonal system is generated by the discretization of an implicit factorization method. A moving grid system fitted to both the free surface and body surface is generated by an effective cubic spline fitting technique. Two zero-equation turbulence models, namely the Cebeci-Smith model and the Baldwin-Lomax model, are used for turbulent calculations. Numerical simulations are carried out for the free surface viscous flows generated by a submerged hydrofoil and a ship model. Computed results are in reasonable agreement with measurements.     

On the nonlinear interfacial instability of rotating core-annual flow

The interfacial stability of rotating core-annular flows is investigated. The linear and nonlinear effects are considered for the case when the annular region is very thin. Both asymptotic and numerical methods are used to solve the flow in the core and film regions which are coupled by a difference in viscosity and density. The long-time behavior of the fluid-fluid interface is determined by deriving its nonlinear evolution in the form of a modified Kuramoto-Sivashinsky equation. We obtain a generalization of this equation to three dimensions. The flows considered are applicable to a wide array of physical problems where liquid films are used to lubricate higher- or lower-viscosity core fluids, for which a concentric arrangement is desired. Linearized solutions show that the effects of density and viscosity stratification are crucial to the stability of the interface. Rotation generally destabilizes nonaxisymmetric disturbances to the interface, whereas the centripetal forces tend to stabilize flows in which the film contains the heavier fluid. Nonlinear effects allow finite-amplitude helically traveling waves to exist when the fluids have different viscosities.This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. This work was also supported by the Science and Engineering Research Council.     

Original Article On the frame dependence and form invariance of the transport equations for the Reynolds stress tensor and the turbulent heat flux vector: its consequences on closure models in turbulence modelling

The present paper shows that the transport equations governing second order turbulent closures are form invariant, but remain frame dependent through the emergence of the body force; thus they do not fulfil the principle of material frame indifference as formulated by Truesdell & Noll (1965). However, this frame dependence corresponds to that first discussed by Müller (1972) and today developed in the framework of the new concept of extended thermodynamics. Following this new concept, these relations are consequently incorporated as additional basic balance laws. The results are: 1) in the case of the Reynolds-stress-transport equation, this eliminates the so-called constraints imposed in [15–17, 19] on turbulence models; 2) to ensure the closure of the new set of basic balance laws, closure assumptions can then be considered as proper constitutive equations which must be restricted by the well known constitutive theory principles in extended thermodynamics. Received: April 4, 1996     

Numerical study of the noninertial systems: application to train coupler systems

Car coupler forces have a significant effect on the longitudinal train dynamics and stability. Because the coupler inertia is relatively small in comparison with the car inertia; the high stiffness associated with the coupler components can lead to high frequencies that adversely impact the computational efficiency of train models. The objective of this investigation is to study the effect of the coupler inertia on the train dynamics and on the computational efficiency as measured by the simulation time. To this end, two different models are developed for the car couplers; one model, called the inertial coupler model, includes the effect of the coupler inertia, while in the other model, called the noninertial model, the effect of the coupler inertia is neglected. Both inertial and noninertial coupler models used in this investigation are assumed to have the same coupler kinematic degrees of freedom that capture geometric nonlinearities and allow for the relative translation of the draft gears and end of car cushioning (EOC) devices as well as the relative rotation of the coupler shank. In both models, the coupler kinematic equations are expressed in terms of the car body and coupler coordinates. Both the inertial and noninertial models used in this study lead to a system of differential and algebraic equations that are solved simultaneously in order to determine the coordinates of the cars and couplers. In the case of the inertial model, the coupler kinematics is described using the absolute Cartesian coordinates, and the algebraic equations describe the kinematic constraints imposed on the motion of the system. In this case of the inertial model, the constraint equations are satisfied at the position, velocity, and acceleration levels. In the case of the noninertial model, the equations of motion are developed using the relative joint coordinates, thereby eliminating systematically the algebraic equations that represent the kinematic constraints. A quasistatic force analysis is used to determine a set of coupler nonlinear force algebraic equations for a given car configuration. These nonlinear force algebraic equations are solved iteratively to determine the coupler noninertial coordinates which enter into the formulation of the equations of motion of the train cars. The results obtained in this study showed that the neglect of the coupler inertia eliminates high frequency oscillations that can negatively impact the computational efficiency. The effect of these high frequencies that are attributed to the coupler inertia on the simulation time is examined using frequency and eigenvalue analyses. While the neglect of the coupler inertia leads, as demonstrated in this investigation, to a much more efficient model, the results obtained using the inertial and noninertial coupler models show good agreement, demonstrating that the coupler inertia can be neglected without having an adverse effect on the accuracy of the solution.     

Some developments in computational modeling of turbulent flows

In this paper, some recent developments of two turbulence closure schemes at ICOMP, NASA Lewis will be discussed. One is the Reynolds-stress algebraic equation model and the other is the Reynolds-stress transport equation model. Various model constraints required by the rapid distortion theory, the invariant theory and the realizability principle, etc. will be described in the model development. The models discussed are for high-turbulent Reynolds number flows, so that the near-wall turbulence and the low-Reynolds-number turbulence are not discussed here.     

An application of Roe's flux-difference splitting for k-ϵ turbulence model

In this paper Roe's flux-difference splitting is applied for the solution of Reynolds-averaged Navier-Stokes equations. Turbulence is modelled using a low-Reynolds number form of the k-? tubulence model. The coupling between the turbulence kinetic energy equation and the inviscid part of the flow equations is taken into account. The equations are solved with a diagonally dominant alternating direction implicit (DDADI) factorized implicit time integration method. A multigrid algorithm is used to accelerate the convergence. To improve the stability some modifications are needed in comparison with the application of an algebraic turbulence model. The developed method is applied to three different test cases. These cases show the efficiency of the algorithm, but the results are only marginally better than those obtained with algebraic models.     

Assessment of a high-order accurate Discontinuous Galerkin method for turbomachinery flows

In this work the capabilities of a high-order Discontinuous Galerkin (DG) method applied to the computation of turbomachinery flows are investigated. The Reynolds averaged Navier–Stokes equations coupled with the two equations k-ω turbulence model are solved to predict the flow features, either in a fixed or rotating reference frame, to simulate the fluid flow around bodies that operate under an imposed steady rotation. To ensure, by design, the positivity of all thermodynamic variables at a discrete level, a set of primitive variables based on pressure and temperature logarithms is used. The flow fields through the MTU T106A low-pressure turbine cascade and the NASA Rotor 37 axial compressor have been computed up to fourth-order of accuracy and compared to the experimental and numerical data available in the literature.