A numerical program for dealing with finite-amptlitude disturbances in plane parallel laminar flows

Summary For plane flows the behaviour of finite-amplitude disturbances, periodic in the direction of motion, is studied here in a way which is somehow similar to a numerical simulation. In fact the foundamental hypothesis which in the linear approach conducts for transverse disturbances to the well known Örr-Sommerfeld system of equations — namely that the periodic disturbance depends on time both by an exponential factor for the amplitude and a constant velocity for the phase — is not made here.This requires to solve a system of non-linear partial differential equations respect both to time and to the space coordinate orthogonal to the wall(s).Boundary conditions are written also for walls wavy in the transverse sense.Solution is found integrating step-by-step with a Runge-Kutta procedure and respect to time a system of ordinary first order differential equations obtained using a finite difference approximation for space derivatives.The method allows to study also disturbances far from the neutral ones and to have detailed information on their interaction with main flow.As an example plane Poiseuille flow at a low Reynolds number is considered.

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Sommario Il comportamento delle perturbazioni periodiche di ampiezza finita nei moti laminari piani viene studiatò con un metodo numerico che somiglia ad una simulazione.Sostanzialmente non viene fatta l'ipotesi che le perturbazioni dipendano dal tempo tramite una velocità di fase costante ed una ampiezza di tipo esponenziale; ipotesi che conduce nel caso lineare alle note equazioni di Örr-Sommerfeld.Si deve così risolvere un sistema di equazioni non lineari alle derivate parziali sia rispetto al tempo che alla coordinata ortogonalle alla parete con condizioni sulle pareti che sono state generalizzate per pareti ondulate con scanalature ortogonali al moto, anche dipendenti dal tempo.Le equazioni del moto si riducono ad un sistema ordinario di equazioni differenziali se si esprimono le derivate parziali tramite delle differenze finite. La loro integrazione rispetto al tempo è fatta col metodo di Runge Kutta-Gill del quarto ordine.Si possono così studiare anche perturbazioni lontane dalla curva di neutralità. Come esempio si studia il flusso piano di Poiseuille ad un basso numero di Reynolds.

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This work has been performed within the research group no. 7 of Consiglio Nazionale delle Ricerche.     

Non linear transverse disturbances in plane poiseuille flow at low Reynolds numbers — Part I

Summary The behaviour and sign of the Reynolds stress for periodic perturbation of finite amplitude is studied in this paper for the plane Poiseuille flow. The case considered has Reynolds number 250 and wave number =1. The Reynolds stress that in the linear case was of opposite sign with respect to the viscosity, in the case considered becomes such for perturbation amplitudes which are still significant with respect to the dynamics of the mean flow. Some numerical results are given to characterize the phenomenon.

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Sommario In questo lavoro si studia nel moto piano di Poiseuille il comportamento ed il segno dello stress di Reynolds per perturbazioni periodiche di ampiezza finita. E' trattato il caso del numero di Reynolds 250 e del numero d'onda =1. Si osserva che lo stress di Reynolds, che nel caso lineare era di segno opposto alla viscosità, diventa tale per ampiezza della perturbazione rilevante rispetto alla dinamica del moto medio. Sono dati alcuni risultati numerici che caratterizzano il fenomeno.

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This work is part of a research program on hydrodynamics at the Istituto di Elaborazione dell'Informazione of the C.N.R., Pisa. The first Author has suggested and precisely formulated the problem, whereas the numerical work was carried out by the second Author.     

The thermoconvective instability of plane poiseuille flow heated from below: A proposed benchmark solution for open boundary flows

The transient two-dimensional Navier-Stokes and energy equations have been solved numerically for flow in a horizontal channel heated from below in the Boussinesq limit. For the set of dimensionless parameters chosen, the flow consists of periodic transverse travelling waves resulting from a convective instability. The solution is proposed as a benchmark for the application of outflow boundary conditions (OBC) in time-dependent flows with strong buoyancy effects. Richardson extrapolation in both time and space is used in obtaining the solution. Field plots and profiles of velocity, temperature, vorticity and streamfunction at selected axial positions and times are also presented from the finest grid and smallest time step calculation. The calculations have been made on an extended domain so that the effects of OBC used in the present study would be negligible in the test region.     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

On the dispersion coefficients for poiseuille flow in a circular cylinder

For axisymmetric dispersion in laminar flow in a cylinder of circular cross section, we report the full time dependence of the first three dispersion coefficients, viz. X ₀, X ₁ and X ₂. We investigate the use of a variety of transverse averages. We also indicate the dependence of the dispersion coefficients on the initial solute distribution and on the chemical activity of the solute.     

Non-modal instabilities of two-dimensional disturbances in plane Couette flow of a power-law fluid

Instabilities of fluid flows have traditionally been investigated by normal mode analysis, i.e. by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem. However, the results of eigenvalue analysis agree poorly in many cases with experiments, especially for shear flows. In this paper we study the instabilities of two-dimensional Couette flow of a polymeric fluid in the framework of non-modal stability theory rather than normal mode analysis. A power-law model is used to describe the polymeric liquid. We focus on the response to external excitations and initial conditions by examining the pseudospectra structures and the transient energy growths. For both Newtonian and non-Newtonian flows, the results show that there can be a rather large transient growth even though the linear operator of Couette flow has no unstable eigenvalue. The effects of non-Newtonian viscosity on the transient behaviors are examined in this study. The results show that the “shear-thinning/shear-thickening” effect increases/decreases the amplitude of responses to external excitations and initial conditions.     

A numerical study of riblet effects on laminar flow through a plane channel

Numerical computations are reported of fully-developed flow through plane channels with knife-edge riblets. The study has covered riblet height: spacing ratios from 0.5 to 3 and riblet heights ranging from 0.1%–4.0% of the distance between the channel walls. In no case did results indicate a reduction in drag compared with the case of a smooth channel, a result that is in contrast with earlier studies. It is suggested that apparent drag reductions reported earlier in laminar flow may have arisen from the use of an insufficiently fine grid.     

The effect of longitudinal vibration on poiseuille flow in a non-circular pipe

The effect is discussed of a superposed longitudinal sinusoidal vibration on the flow, under a constant pressure gradient, of a slightly non-Newtonian fluid in a straight pipe of non-circular cross-section. It is found that a flow in transverse planes is produced. This is the superposition of steady and sinusoidal flows. The stream-function corresponding to the steady flow is obtained in the case when the pipe has a rectangular cross-section. The discussion is based on a Rivlin-Ericksen constitutive equation in which only“second-order” non-Newtonian terms are present.     

Two fundamental modes of disturbances and their development in a plane plume above a horizontal heat source

Experimental investigations are conducted on development of the disturbances which appear in the transition to turbulence in a natural convection plume above a horizontal line heat source in air. Both the power spectra of velocity and temperature in natural transition show that there seem to be two fundamental modes of disturbances. One is an outstanding peak about 0.8 Hz and the other a small one about 1.1 Hz in the spectra. The disturbances of these fundamental frequencies are observed as anti-symmetric modes around the entrance to the transition region. The disturbance of the first fundamental frequency is a selectively amplified anti-symmetric mode in that area. In contrast, the disturbance of the second fundamental frequency is thought to be originated from a symmetric mode and then transformed into an anti-symmetric mode of the same frequency during its growth.     

Natural oscillations arising from loss of stability in parallel flows of a viscous liquid under long-wavelength periodic disturbances

It was shown in [1] that a parallel flow with an arbitrary nonconstant velocity profile is unstable for long-wavelength spatially periodic disturbances along the flow. The present paper shows that this instability leads to a supercritical natural oscillation mode of the simple wave type. This mode is calculated using the Lyapunov-Schmidt method in the form given in [2], along with the asymptotic curve of the wavelengths [1]. If the long wavelength disturbances are the most dangerous (this occurs, for example, when there is a sinusoidal velocity profile), then the natural oscillation mode is stable for spatially periodic disturbances having the same wavelength.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 32–35, January–February, 1973.     

Direct numerical simulation of the amplification of a 2D temporal disturbance in plane Poiseuille flow

A direct numerical scheme is developed to study the temporal amplification of a 2D disturbance in plane Poiseuille flow. The transient non-linear Navier–Stokes equations are applied in a region of wavelength moving with the wave propagation speed. The complex amplitude involved in the perturbation functions is considered as the initial input of the non-linear stability equations. In this study a fully implicit finite difference scheme with five points in the flow direction and three points in the normal direction is developed so that numerical simulation of the amplification of a two-dimensional temporal disturbance in plane Poiseuille flow can be investigated. The growth and decay of the disturbance with time are presented and neutral stability curves which are in good agreement with existing solutions can be determined. The critical conditions as a function of the amplitude A₀ of the disturbance are presented. Fixing the wavelength, the Navier–Stokes equations are solved up to Re=10,000 a friction factor increasing with Reynolds number is observed. The 2D non-linear behaviour of the streamfunction, vorticity and velocity components at Re=10,000 are also exhibited. © 1998 John Wiley & Sons, Ltd.     

Molecular model and flow calculation: I. The numerical solutions to multibead-rod models in homogeneous flows

The Galerkin method is used to solve the diffusion equation of the distribution function in configurational space for a multibead-rod model, and the dimensionless components of the extra stress tensor are then calculated by means of the expression of ensemble average. The material functions for steady-state shear flow and uniaxial flow and the mechanical properties of rigid-rodlike molecule suspensions in superposed flows are obtained numerically. The results indicate that it is promising to employ the multibead-rod models without the constitutive equation in numerical simulations of flows of suspensions. The project supported by the National Natural Science Fundation of China.     

Effect of small-amplitude vibrations on viscoelastic flows in a plane slider bearing

The qualitative changes of dynamic lift and friction forces caused by small-amplitude harmonic vibrations superimposed on flows in a plane slider bearing are considered for simple viscous and viscoelastic lubricating fluids. Low- and high-frequency disturbances are analysed in greater detail and the most beneficial situations discussed.     

Spatial direct numerical simulation of high-speed boundary-layer flows Part II: Transition on a cone in Mach 8 flow

The laminar breakdown of the boundary-layer flow of an axisymmetric sharp cone in a Mach 8 flow is simulated by a synergistic approach that combines the parabolized stability equation (PSE) method and spatial direct numerical simulation (DNS). The transitional state is triggered by a symmetric pair of oblique second-mode disturbances whose nonlinear interactions generate strong streamwise vorticity, which leads in turn to severe spanwise variations in the flow and eventual laminar breakdown. The PSE method is used to compute the weakly and moderately nonlinear initial stages of the transition process and, thereby, to derive a harmonically rich inflow condition for the DNS. The strongly nonlinear and laminar-breakdown stages of transition are then computed by well-resolved DNS, with a highly accurate algorithm that exploits spectral collocation and high-order compact-difference methods. Evolution of the flow is presented in terms of modal energies, mean quantities (e.g., skin friction), Reynolds stresses, turbulent kinetic energy, and flow visualization. The numerical test case is an approximate computational analog of one of the few stability experiments performed for hypersonic boundary-layer flows. Comparisons and contrasts are drawn between the experimental and the computational results. Rope-like waves similar to those observed in schlieren images of high-speed transitional flows are also observed in the numerical experiment and are shown to be visual manifestations of second-mode instability waves.This research was supported under NASA Contracts NAS1-19831 and NAS1-20059 for the first and second authors, respectively.     

平面弹性介质多孔多裂纹问题的一种数值方法

提出了平面弹性介质中多孔洞多裂纹相互作用问题的一种数值计算方 法. 通过把适于单一裂纹的Bueckner原理扩充到含有多孔洞多裂纹的一般体系，将原问题 分解为承受远处载荷不含裂纹不含孔洞的均匀问题，和在远处不承受载荷但在裂纹面上和孔 洞表面上承受面力的多孔洞多裂纹问题. 于是，以应力强度因子作为参量的问题可以通过考 虑后者(多孔洞多裂纹问题)来解决，而利用提出的杂交位移不连续法，这种多孔 洞多裂纹问题是容易数值求解的. 算例说明该数值方法对分析平面弹性介质中多孔洞多裂纹 相互作用的问题既简单又有效.     

A theorem for a shear stress-free plane boundary in Stokes flow

A theorem for non-axisymmetrical Stokes flow about a shear stress-free plane boundary is established by using a representation for the velocity and pressure fields for the same flow in terms of biharmonic and harmonic functions. A corollary of the theorem is derived which gives the axisymmetrical Stokes flow in terms of the Stokes function about the same boundary. The formulae for drag and torque on the boundary are also given. A few illustrative examples are presented.     

Leading-edge flow criticality as a governing factor in leading-edge vortex initiation in unsteady airfoil flows

A leading-edge suction parameter (LESP) that is derived from potential flow theory as a measure of suction at the airfoil leading edge is used to study initiation of leading-edge vortex (LEV) formation in this article. The LESP hypothesis is presented, which states that LEV formation in unsteady flows for specified airfoil shape and Reynolds number occurs at a critical constant value of LESP, regardless of motion kinematics. This hypothesis is tested and validated against a large set of data from CFD and experimental studies of flows with LEV formation. The hypothesis is seen to hold except in cases with slow-rate kinematics which evince significant trailing-edge separation (which refers here to separation leading to reversed flow on the aft portion of the upper surface), thereby establishing the envelope of validity. The implication is that the critical LESP value for an airfoil–Reynolds number combination may be calibrated using CFD or experiment for just one motion and then employed to predict LEV initiation for any other (fast-rate) motion. It is also shown that the LESP concept may be used in an inverse mode to generate motion kinematics that would either prevent LEV formation or trigger the same as per aerodynamic requirements.