A Dissipation Rate Equation for Low-Reynolds-Number and Near-Wall Turbulence

Two-equation turbulence models are usually formulated for specific flow types and are seldom validated against a variety of flows to account for near-wall and low-Reynolds-number effects simultaneously. In addition to low-Reynolds-number effects, near-wall flows also experience wall blocking, which is absent in free flows. Consequently, near-wall modifications to two-equation models could be quite different from low-Reynolds-number corrections. Besides, it is known that existing two-equation models perform poorly when used to calculate plane wall jets and two-dimensional backstep flows. These problems could be traced to the modeling of the dissipation rate equation. In this paper an attempt is made to improve the modeling of the dissipation rate equation so that it could successfully predict both free and wall-bounded shear flows including plane wall jets and backstep flows. The predictions are compared with experimental and direct numerical simulation data whenever available. Most of the data used are obtained at low Reynolds numbers. Good correlation with data is obtained. Therefore, for the first time, a model capable of correctly predicting free and wall-bounded shear flows, backstep flows, and plane wall jets is available. Received: 12 December 1995 and accepted 12 November 1996     

Supersonic Inviscid Corner Flows with Regular and Irregular Shock Interaction

The results of a numerical and analytical investigation of steady-state supersonic inviscid flows in corners formed by intersecting compression wedges are presented. The flows considered are symmetric about the corner bisector. The distinctive features of flow pattern formation related with the reflection of wedge-generated shocks from the bisector plane are studied. The wedge angles at which transition from regular to irregular shock reflection occurs are determined both numerically and analytically using the criteria available for plane flows; the data thus obtained are found to be in agreement. Flow patterns with irregular shock reflection, namely, single, transitional, and double Mach, as well as von Neumann reflection, are identified; they are similar to the known types of reflection for plane quasi-steady-state flows. Varieties of these types not observed in the plane flows are found to exist. The effects of the angle of inclination of the plane surfaces of the corner to the freestream direction, the sweep angle of the leading edges, and the dihedral angle are investigated. Some previously unknown parameters of corner configurations for which transition may occur in accordance with the von Neumann criterion are determined.     

Axisymmetric mixed flows in magnetogasdynamics

In contrast with conventional gasdynamics, in magnetogasdynamics there are several types of mixed flows. A detailed study of such plane flows was first made by Kogan [1]. After this, intensive work was done on the magnetogasdynamic mixed flows [2–13], with the plane case being considered in all the studies except [9]. In [9] the equations of the possible mixed flows for the axisymmetric case were obtained in terms of the disturbance velocity components.The axisymmetric mixed flows are studied in detail in the present paper. The exact equations of motion are obtained for the velocity potential and the streamfunction, and the corresponding approximate equations are obtained for all the transitional regimes (transonic, hypercritical, trans-Alfvenic, transonic-trans-Alfvenic). Simple particular solutions are obtained for these approximate equations.For greater generality the entire study is made simultaneously for the plane and axisymmetric cases.The author wishes to thank S. V. Fal'kovich for his interest in the study and for valuable discussions.     

Topological visualisation of focal structures in free shear flows

This paper describes a method for identifying and visualising the three-dimensional geometry of focal (vortex) structures in complex flows. The method is based primarily on the classification of the local topology as it is identified from the values of the velocity gradient tensor invariants. The identification of the local topology is reference frame invariant. Therefore, focal (vortex) structures can be unambiguously identified in these flows. A novel flow visualisation method is introduced whereby focal structures are rendered using a solid model view of the local topology. This new approach is applied to the identification of focal structures in three-dimensional plane mixing layer and plane wake flows.     

Approximate two-dimensional equations of vortex flow in a plane layer with solid boundaries

Approximate two-dimensional equations governing turbulent vortex flows in plane fluid layers are considered. The equations were derived by the author in his earlier studies using the shallow water approximation and neglecting circulatory flows in the layer cross-sections. It is shown that, due to the centrifugal effect in the vortex flow, return flows in the layer cross-sections have only a slight influence on the fluid flow in the plane layer and can be neglected.     

Hodograph transformation in steady plane rotating MHD flows

A hodograph transformation is employed to obtain a partial differential equation of second order which is exploited to obtain solutions for plane rotating viscous incompressible flows with orthogonal magnetic and velocity fields. Lastly radial and vortex flows are discussed.     

Les mouvements par nappes ou tranches planes et parallèles d'un fluide visqueux incompressible

Here we make use of the vocable ‘slices or layer-type’ (for lack of better appellation) whenever a pencil of plane parallel fluid sections is so conveyed by the flow that plane remain the sections. Pseudo stream functions and general equations governing the flow are proposed so as to enlarge the theory of plane and pseudo-plane flows. By taking the aspects of solid boundaries into account, we show that the layer-type flows can be sharply divided into three categories. Then by seeking polynomial solutions of the equations, we reveal two families of exact solutions of Navier–Stokes equations.     

Effects of a transverse flows with variable viscosity in micropolar fluids

A regular perturbation analysis is presented for three laminar natural convection flows in micropolar fluids in liquids with temperature dependent viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface. While these flows have well-known power-low similarity solutions when the fluid viscosity is taken to be constant, they are non-similar when the viscosity is considered to a function of temperature. A single similar flow, that adjacent to a vertical isothermal surface, is also analysed for comparison in order to estimate the extent of validity of perturbation analysis. The formulation used here provides a unified treatment of variable viscosity effects on those four flows. Computed first-order perturbation quantities are presented for all four flows. Numerical results for velocity, angular velocity and thermal functions has been shown graphically or tabulated for different values of micropolar parameters. Received on 20 October 1997     

具有热源项的驻点流动和传热问题的近似解析解

研究持续拉伸变形表面上二维平面和轴对称驻点流的动量和热量传输问题。利用同伦分析方法获得速度分布和温度分布的级数解,讨论了级数的敛散性。通过图形分析主流速度与拉伸速度的比率参数,普朗特参数,热源参数和流动类型指标对速度边界层和温度边界层的影响。结果表明,这些参数对二维平面驻点流动和传热有较大的影响。     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

Onset of shear-layer instability at the interface of parallel Couette flows

A non-planar or a bilateral mixing-layer is studied by means of a series of direct numerical simulations (DNSs). This mixing-layer forms at the interface of two co-current plane Couette flows of different Reynolds numbers. The current DNS study determined the conditions for the onset of shear-layer instability at the interface. The influence of different Reynolds number (of the co-current plane Couette flows) and their Reynolds number ratio on the mixing-layer is studied. A critical Reynolds number of about 500 (or more particularly one of the co-current plane Couette flows must be turbulent) and a Reynolds number ratio greater than 2 is required for the genesis of this bilateral shear-layer instability. Independent of the Reynolds number and the Reynolds number ratio, the temporal evolution of the shear-layer instability followed the same pattern. In addition, the oscillation frequency of the instability was found to increase with increasing Reynolds number and increasing Reynolds number ratio. Further, influence of instability on the local skin friction and the two-point correlation is elaborated on.     

Some superposition theorems for second-order fluids

Theorems are derived, within the framework of the second-order Rivlin-Ericksen theory, relating to the superposition of longitudinal flows on plane flows in an incompressible isotropic viscoelastic fluid. Also, some theorems are obtained concerning the effects of inertia on the solution of plane flow problems in such viscoelastic fluids.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

Three-dimensional shock intersection

The flow field in the neighborhood of the three-dimensional intersection of two shocks of different families is investigated when in the plane perpendicular to the line of intersection the flow velocity component is subsonic behind at least one of the departing shocks. In the plane case these flows are not realized. The boundary of the domain of the key parameters for which these flows are possible is determined. The characteristics of the flow field are determined when: (1) behind the departing shocks the flow is homogeneous, and (2) the velocity vectors behind the departing and arriving shocks are parallel to a single plane which contains the intersection line. The flow in Mach-type shock intersection in the neighborhood of the intersection lines (triple points in the plane) is a particular case of the problem considered. It is shown that Mach-type shock intersection is not possible when the intensity of the arriving shocks is less than for their steady-state Mach intersection in the calculation plane. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 137–143, November–December, 1998.     

Role of the pressure gradient in flows controlled by a near-wall body force

Steady two-dimensional flows of a viscous incompressible fluid over a plane boundary in the presence of a near-wall body force are studied. The calculations are performed for a very simple one-parameter force distribution simulating the action of a single plasma actuator. The important role played by the pressure gradient in the flows controlled by body forces is shown.     

Two-dimensional free boundary layer in a stratified medium

Expressions for the fluctuation characteristics of shear flow in a stratified medium are obtained on the basis of the equations for the single-point second-order moments of the velocity and temperature fields and then closure of those equations by means of semiempirical hypotheses. The Prandtl equation, with the influence or Archimedean forces taken into account, is used to analyze plane jet flows and wake flows of a body, Numerical computations are carried out for a plane wake, and the results are compared with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 71–79, July–August, 1977.     

ON LIAPOUNOFF’S METHOD IN THE THEORY OF STABILITY OF LAMINAR FLUID FLOWS

In[1]Zhou extended some Liapounoff‘s theorems of the theory of stability in the case of plane laminar fluid flows.In[2]Zhou and Li investigated the eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation,and obtained some new results.In this paper,based on the results of[1]and[2]we shall prove:(1)For the linearized problem the definition of stability according to the eigenvalues of Orr-Sommerfeld equation and that according to the perturbation.energy are equivalent;(2)The method of linearization is admissible for the stability pro-blem of plane laminar fluid flows for sufficiently small initial disturbance.     

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the OrrSommerfeld and Squire(OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a reorthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes(N-S) equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is proposed. The algorithm is then applied to a one-dimensional base flow(the plane Poiseuille flow) and a two-dimensional base flow(the plane Poiseuille flow with a low-speed streak)in a channel. For one-dimensional cases, the effects of the spanwise width of the channel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is characterized by high-and low-speed streaks and the corresponding streamwise vortical structures.The lift-up mechanism that induces the transient growth of perturbations is discussed.The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one-and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the crosscheck method.     

Predictive Capabilities of an Improved Cubic k–ε Model for Inert Steady Flows

Through an improved ε transport equation, a major quality enhancement of the cubic k–ε model, earlier developed in[13], is obtained. The ε-equation of [13],yielding good results for wall-bounded and rotating flows, is combined with the one derived by Shih et al. [20], which produces good results for free shear flows (e.g. the plane jet–round jet anomaly is resolved).Results are presented for the following flows: fully developed stationary and rotating channel and pipe, backward-facing step, sudden pipe expansion, smooth channel expansion and contraction, plane and round jet. Heat transfer predictions in turbulent impinging jets are also discussed. Accurate results are obtained for the mean flow quantities for all test cases, without case dependent model tuning. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some exact solutions of the equations of a stationary monoenergetic beam of charged particles

We consider the class of invariant solutions which can describe only vortex flows (curl P 0, P is the generalized momentum) and show that they contain solutions corresponding to flows from a plane or cylindrical emitter with a linear voltage drop across it (direct heating) in the temperature-limited regime^*. The solution is obtained in analytic form for emission from a plane in a uniform magnetic field perpendicular to the flow plane. It also (for=0) defines a plane magnetron in the T-regime. The solution of the problem for a cylindrical emitter reduces to considering equations describing a cylindrical diode or magnetron in the T-regime, where the shape of the collector is given by the potential distribution curve for these cases. We can extend the results to a relativistic beam if restrictions are imposed on its relative dimensions which permit us to ignore the magnetic self-field. Brillouin type flows (including irrotational ones) are studied in which particles move without intersecting the equipotential surfaces along three-dimensional spirals on the surface of cones. An analytic solution is given for relativistic Brillouin flow in a conical diode when strict allowance is mede for the magnetic self-field.