Analysis of PSE for Perturbed Flows_presentation

8/8/2019 Analysis of PSE for Perturbed Flows_presentation

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By:

Nishant Kumar

Supervisor:

Dr. Vinod Narayanan

INDIAN INSTITUTE OF TECHNOLOGY

Gandhinagar

ANALYSIS OF PARABOLIZED STABILITY

EQUATIONS (PSE) FOR PERTURBED FLOWS


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Contents

About

Background

Navier Stokes Equations

Orr Sommerfeld Equations (OSE)

Solution of Orr-Sommerfeld equation in cylindrical equations

Parabolized Stability Equations (PSE) for Fluid Flow

Development

Usage

2D-PSE for Incompressible Flow

PSE Concept in Natural Variables

Closure and Norm

Solution Procedure

Conclusion

References

Analysis of PSE for PerturbedFlows

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About

The parabolized stability equations (PSE) are a new approach to analyse thestreamwise evolution of single or interacting Fourier modes in weaklynonparallel flows such as boundary layers.

The concept rests on the decomposition of every mode into a slowly varyingamplitude function and a wave function with slowly varying wave number.

The neglect of the small second derivatives of the slowly varying functions with respect to the streamwise variable leads to an initial boundary-valueproblem that is solved by numerical marching procedures.

The PSE approach is valid in convectively unstable flows.

The PSE codes have developed into a convenient tool to analyse basic

mechanisms in boundary-layer flow.Here, the emphasis has been laid on the analysis of PSE concept in natural variables, along with the underlying background and concepts. Theimplementation of the equations using FORTRAN codes was done to analysethe solutions of the equations.


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Background

Some theoretical background is required for understanding the conceptof hydrodynamic stability and analysis of fluid flows.


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Navier Stokes Equations

The NSE, along with the continuity equation and well formulated boundary conditions,

is used to determine the flow field. The NSE is basically a statement of conservation ofmomentum. For incompressible flow of Newtonian fluids, the NSE in vector form maybe represented as

The work deals basically with the axisymmetric flows and hence it is beneficial to representthe N-S equation in cylindrical coordinates systems in non-dimensional form. A change invariables of the cartesian form of the equation will yield the desired form in cylindricalcoordinates (r,,z).


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N-S equations in cylindrical co-ordinates


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Orr Sommerfeld Equation (OSE)

Amplification of disturbances: The flow may consist of disturbances caused due to freestream condition or surface roughness. Disturbances of a certain band of frequenciesget damped by the fluid viscosity and some frequencies get amplified, which may causeturbulence downstream of the flow.

The OSE is an eigenvalue equation describing the effect of linear two-dimensionalmodes of disturbance on a viscous parallel flow. The perturbed velocity field is

where perturbed velocity component is determined from


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In non-dimensional form, OSE can be obtained from the linearized version of NSE. Thefinal form of OSE can be represented as

Solution of OSE: For the OSE to give non-trivial solution, we must have

It is assumed that the disturbances are temporally growing. Thus, depending on the valueof c, the -Re plane can be divided into following domains

c < 0, flow is stable (i.e., disturbances decay),

c = 0, flow is neutrally stable, and

c > 0, flow is unstable (i.e., disturbances grow).


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Fig General shape of the neutral stability curve (locus of ci =0 in -Re space) for a Blasiusboundary layer and for layers with adverse pressure gradients.


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Solution of Orr-Sommerfeld equation in

cylindrical co-ordinates

For axisymmetric flows, to investigate the stability of the basic flow, we study theevolutions of the perturbations u, vandp requiring that

total flow parameters satisfy the equations of motion and

they are constrained by the same flow-forcing boundary conditions as the basic flow.

Here, we consider the perturbed flow to be given as the combination of the unperturbed

mean flowV,Pand the perturbations v,p

The disturbances are also assumed to be parallel flows. Then the most general form forthe 1-D disturbance q is that of a travelling wave

whose amplitude varies with r, and

which moves along the boundary at an angle with respect to the zaxis.

In complex notation, the disturbances may be represented as

All disturbances have wave number and frequency. They are referred to as Tollmein-Schlichting waves which are indications of laminar-flow instability.


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Now, we substitute the perturbed velocity components given into the N-S equation andsubtract out the basic flow equalities and neglect higher powers and product of p. Thelinearized disturbance equations may then be written as

We eliminate the terms containingPfrom the r-zand-zequations to obtain two equations

with u,v,was the unknowns.We eliminate the axial velocity term from the above equationsby using the continuity equation, and by incorporating the exponential disturbanceequation, we get

where, dash represents the differentiation with respect to r.


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The equation has three parameters: , c and .

For a given profile, only a certain continuous but limited sequence of these parametersthe eigenvalues will satisfy the relation.

The mathematical problem is to find this sequence, which has a different functionalform for spatial versus temporal growth of disturbances.

A FORTRAN program that takes the velocity profile as input and evaluates thecorresponding eigenvalues from the relation given by above eqn. was used to analyse theflow for stability.


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Parabolized Stability Equations (PSE)

for Fluid Flow

Development

The laminar-turbulent transition of boundary layer flows led to the linear stability theory,assuming parallel flow. The resulting eigenvalue problem, expressed as Orr-Sommerfeld(OSE) and Squire equations, when used for boundary layer transition, showed deviationsfrom the experimental results for the neutral curve. These discrepancies were attributed to

the local assumption of parallel flow and

the consideration of no upstream development of boundary layer.

Consequently, this led to a more general analysis that takes into account both

the streamwise development and

upstream data of the instability.

This analysis further led to the PSEs, applicable to 2D-flows (2D-PSE). These equationsare valid for convective instabilities and relax the assumption of parallel flow.


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Usage

The PSEs are used to analyse the streamwise evolution of Fourier modes in weakly non-parallel flows.

It relies on the decomposition of each mode into a slowly varying amplitude functionand a wave function with slowly varying wave number.

Analogous to the OSE, several stability analyses generalise the flow assumptions in 2Dplane. This concept is inadequate in prediction of breakdown as it neglects the axialgradients that may influence the vortex.

This necessitates an analysis technique that can account for velocity and pressuregradients.

Thus, the 3D-PSE concept originated which has an advantage of being able to solve forarbitrarily complicated basic states in the x,yplane.


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2D-PSE for Incompressible Flow

The continuity equation and non-linear momentum equation for an incompressible fluidwith constant density and viscosity are given by

We consider the stability of 2D steady laminar boundary-layer flows neglecting

the mean flow, and

derivatives in the spanwise zdirection.

For stability analysis, we decompose the total flow field into the steady laminar basic flowV,P and the disturbancev,p.


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Under boundary-layer approximation, the basic flow is governed by

For disturbancesv,p we obtain the linearized stability equations

For a given basic flow, this system is equivalent to the Navier-Stokes equations and henceof elliptic type.


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PSE Concept in Natural Variables

We maintain the decomposition of the disturbance q into an amplitude function q and awave function and write the disturbance in the form

By substituting the components of above equations into the linearized equations, weobtain in nondimensional form

where the operators L,MandNact only inyand are given as follows (In these equations,D represents differentiation wrty)


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The derivation of above equation involves the following steps:

The elimination of the pressure by taking the curl of the momentum equations whichprovides the vorticity transport equations.

The Orr-Sommerfeld equation is obtained by subtracting the z derivative of the x-vorticity equation from the xderivative of the z-vorticity equation and eliminatingw,

via continuity equation.

For a proper function (x),

the streamwise variation ofq is governed by the wave function and

the derivatives qx and x are small.

The PSE approximation assumes the variation of q and as sufficiently small to neglectqxx, xx, xqx and their higher derivatives with respect to x. The final equation derivedabove is obtained after application of this approximation.


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Closure and Norm

Solving the PSE as an initial boundary value problem is difficult due to the appearance ofd/dx which affects the operators. In absence of any information regarding thestreamwise changes, we can impose the restriction x=0and absorb all streamwise changesinto the amplitude function.

Alternatively, a streamwise independent norm can be imposed on q. Such anorm can be applied at a fixed position given by the maximum streamwise rms fluctuationof the function |u|. The deformation of this function may lead to jumps in value oflocation. We, thus, prefer an integral norm. For defining in the domain , we use

where the dagger denotes the complex conjugate. We now choose

to normalize q.


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The stability equation derived above along with one of the conditions given above, or anysuitable norm are the closed system of parabolized stability equations (PSE). This systempermits the simultaneous calculation of (x) and q(x,y) in a streamwise marchingprocedure.

Different norms lead to different partitions of the solution q. The physicalsolution q, however, is the same to within small effects of the norm on the PSE

approximation.


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Solution ProcedureBefore solving the PSE, we need to specify initial and boundary conditions as follows:

specify Re, , andV,

provide the same estimates 0 for the eigenvalue and q0 for the eigenfunction as initialconditions at xs, and

apply the homogeneous boundary conditions. To solve the PSE, the differential operators and boundary conditions are

converted into algebraic form by any method suitable for the Orr-Sommerfeld problem.We use a spectral collocation method with Chebyshev polynomials, together with a domaintransformation.

Since appears in the differential operators, the system is nonlinear, and a

predictor-corrector approach is employed to find the solution at j+1 (j is the step-index).

where n is the iteration count.


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We exploit the following equation to obtain an updated wavenumber

The integration-update cycle is continued until

the norm is satisfied within a given error limit, and

the solution has converged to j+1, qj+1(y).

We then proceed to the next step in x where a better estimate for

and

x is known.The marching procedure terminates if the convergence criterion is not satisfied

within a given number of iterations.


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PSE as compared to DNS

The Direct Numerical Simulation (DNS) method analyses the transition process foridealized problems of linear stability theory.

The DNS solution is not yet feasible as an engineering method due to the enormouscomputational costs.

Thus, this constrains the transition simulations in boundary layers to a few runs.

However, PSE takes advantage of more efficient methods available for solvingdifferential equations with marching procedures.

The efficiency of this approach and the good agreement of PSE and DNS solutionsencourage its further development as a tool for transition analysis and prediction.


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From the figure, we find perfect agreement between the PSE and DNS.

FigAmplitude growth curves. Comparison of PSE and DNS.


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Work Ahead

The code for evaluating the most unstable mode of disturbance is being developed toobtain the coefficient matrices for the PSE equations.

The axisymmetric flow over a cylinder is to be analysed.

The growth curve for different modes of disturbances and their streamwise evolutionis to be evaluated for laminar to turbulent transition.


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Conclusion

The parabolized stability equations have developed into a matureapproach to analyse the receptivity, linear, and nonlinear stability ofconvectively unstable flows into the late stages of transition.

The applications of the PSE approach for transition analysis in

aerodynamic flows aim towards disturbance environment and locationof the transition point based on the nonlinear changes in skin friction.

The methods discussed here in the report will be useful in betterdocumentation of the disturbance environment.


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References

[1] Michael S. Broadhurst and Spencer J. Sherwin. The parabolised stability equations for 3d- flows: Implementation and numerical stability.Applied Numerical Mathematics, 58:1017-1029,2008.

[2] Thorwald Herbert. Parabolized stability equations. Special Course on Progress inTransition Modelling, AGARD REPORT 793:4/1-34, 1994.

[3] Thorwald Herbert. Parabolized stability equations. Annual Review of Fluid Mechanics,29:245-283, 1997.

[4] Fei Li and Mujeeb R. Malik. Spectral analysis of parabolized stability equations. Computers &Fluids, Vol. 26:pp. 279-297, 1997.

[5] S.S. Motsa and P. Sibanda. On the Chebyshev Spectral Collocation method in channel and jetflows. Journal of Pure Applied Mathematics, 1(1):36-47, 2001.

[6] Helen L. Reed. Parabolized stability equations: 2-d flows. NATO, RTO-EN-AVT-151:4/1-26.

[7] Vassilios Theofilis. Advances in global linear instability analysis of nonparallel and three-dimensional flows. Progress in Aerospace Sciences, 39:249-315, 2003.


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