The Stability of Laminar Flows P M V

The Stability of Laminar Flows P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A Class of Laminar flows have a fatal weakness … What is this Weakness? ? ?

Wake flow of a Flat Plate

Boundary layer development along a wedge We know that the velocity distribution outside the boundary layer is a simple power function

ODE for Wedge BL Flows Introducing the same relationship for the stream function as This is the so-called Falkner-Skan equation which describes a flow past a wedge.

Effect of Wedge Angle on BL Profile

Laminar Jet Flows

Mixing Layer Flow

The Philosophy of Instability • The equations of Fluid dynamics allow some Velocity Profiles. • Given a Velocity Profile, is it stable? If the flow is disturbed, will the disturbance gradually die down, or will the disturbance grow such that the flow departs from its initial state and never recovers?

Major Classes of Instability in Fluid Dynamics • Wall-bounded flows: – Boundary layers, pipe flows, etc – Any basic flow without inflexion point – viscosity plays a role – sensitive to the form of the basic flow • Free-shear flows: – mixing layers, wakes, jets, etc – less sensitive to the form of the basic flow – Viscosity is not responsible.

Laminar-Turbulent Transition

Transition process along a flat plate Fluid Mechanics for Engineers, A Graduate Textbook, Meinhard T. Schobeiri, 2010.

Receptivity of Boundary Layer to Disturbances

Sketch of transition process in the boundary layer along a flat Plate A stable laminar flow is established that starts from the leading edge and extends to the point of inception of the unstable two-dimensional Tollmien-Schlichting waves. Fluid Mechanics for Engineers, A Graduate Textbook, Meinhard T. Schobeiri, 2010.

Sketch of transition process in the boundary layer along a flat Plate Onset of the unstable two-dimensional Tollmien-Schlichting waves.

Sketch of transition process in the boundary layer along a flat Plate Development of unstable, three-dimensional waves and the formation of vortex cascades.

Sketch of transition process in the boundary layer along a flat Plate Bursts of turbulence in places with high vorticity.

Sketch of transition process in the boundary layer along a flat Plate Intermittent formation of turbulent spots with high vortical core at intense fluctuation.

Sketch of transition process in the boundary layer along a flat Plate Coalescence of turbulent spots into a fully developed turbulent boundary layer.

Outline of a Typical Stability Analysis • Small disturbances are present in any flow system. • Small-disturbance stability analysis is followed to understand the receptivity of flow. • This analysis is carried-out in seven steps. 1. The flow problem, whose stability is to be studied must have a basic flow solution in terms of Q 0, which may be a scalar or vector function. 2. Add a disturbance variable Q' and substitute (Q 0 + Q') into the basic equations which govern the problem.

Stability of Flow due to Small Disturbances • We consider a steady flow motion, on which a small disturbance is superimposed. • This particular flow is characterized by a constant mean velocity vector field and its corresponding pressure. • We assume that the small disturbances we superimpose on the main flow is inherently unsteady, three dimensional and is described by its vector filed and its pressure disturbance. • The disturbance field is of deterministic nature that is why we denote the disturbances. • Thus, the resulting motion has the velocity vector field: and the pressure field:

NS Equations for (Steady) Incompressible Flow influenced by Small Disturbances Performing the differentiation and multiplication, we arrive at: The small disturbance leading to linear stability theory requires that the nonlinear disturbance terms be neglected. This results in

Step 3 • From the equation(s) resulting from step 2, subtract away the basic terms which Q 0, satisfies identically. • What remains is the Governing Equation for evolution of disturbance s.

Implementation of Step 3 • Above equation is the composition of the main motion flow superimposed by a disturbance. • The velocity vector constitutes the Navier-Stokes solution of the main laminar flow. • Obtain a Disturbance Conservation Equation by taking the difference of above and steady Laminar NS equations

Disturbance Conservation Equation in Cartesian index notation is written as This equation describes the motion of a three-dimensional disturbance field modulated by a steady three-dimensional laminar main flow field. A solution to above equation will be studied to determine the stability of main flow. Two assumptions are made in order to find an analytic solution. The first assumption implies that the main flow is assumed to be two-dimensional, where the velocity vector in streamwise direction changes only in lateral direction

• The second assumption concerns the disturbance field. • In this case, we also assume the disturbance field to be twodimensional too. • The first assumption is considered less controversial, since the experimental verification shows that in an unidirectional flow, the lateral component can be neglected compared with the longitudinal one. • As an example, the boundary layer flow along a flat plate at zero pressure gradient can be regarded as a good approximation. • The second assumption concerning the spatial two dimensionality of the disturbance flow is not quite obvious. • This may raise objections that the disturbances need not be two dimensional at all.

Two-dimensional Disturbance Equations The continuity equation for incompressible flow yields: With above equations there are three-equations to solve three unknowns.

Step 4 • Linearize the disturbance equation by assuming small disturbances, that is, Q' << Q 0 and neglect terms such as Q’ 2 and Q’ 3 ……. .

GDE for Modulation of Disturbance

Disturbance as A Travelling Wave A travelling disturbance is mathematically defined as a complex stream function: is the complex function of disturbance amplitude which is assumed to be a function of y only. The stream function can be decomposed into a real and an imaginary part:

The perturbation Velocity Field The components of the perturbation velocity are obtained from the stream function as: Introduce the disturbance velocities into stability equations:

Step 6 • The linearized disturbance equation should be homogeneous and have homogeneous boundary conditions. • In other words, it is an eigenvalue problem. • It can thus be solved only for certain specific values of the equation's parameters.

Orr-Sommerfeld -equation • The Orr-Sommerfeld -equation was derived by Orr and independently Sommerfeld. • This equation is obtained by Introducing disturbance velocity functions into modulation equations. • Eliminate the pressure terms by differentiating the first component of the equation with respect to y and the second with respect to x respectively and subtracting the results from each other. This constitutes the fundamental differential equation for stability of laminar flows in dimensionless form.

Orr-Sommerfeld Eigen value Problem • The Orr-Sommerfeld equation is a fourth order linear homogeneous ordinary differential equation. • With this equation the linear stability problem has been reduced to an eigenvalue problem. : OSEV Equation contains the main flow velocity distribution U(y) which is specified for the particular flow motion under investigation, the Reynolds number, and the parameters , cr, and ci. cr is the propagation velocity component in the x-direction. This is known as the "phase velocity". ci is the degree of damping. If ci is negative, it is amplification.

Secrets of Stability • The secrets of infinitesimal laminar-flow instability lie within this fourth-order linear homogeneous equation, first derived independently by Orr (1907) and Sommerfeld (1908). • The boundary conditions are that the disturbances u and v must vanish at infinity and at any walls (no slip). • Hence the proper boundary conditions on the Orr-Sommerfeld equation are of the following types: Boundary layers:

Duct flows: Free shear layers:

Step 7 • The eigenvalues found in step 6 are examined to determine when they grow (are unstable), decay (are stable), or remain constant (neutrally stable). • Typically the analysis ends with a chart showing regions of stability separated from unstable regions by the neutral curves.

Orr-Sommerfeld Eigen Value Problem • Orr-Sommerfeld Equation contains the main flow velocity distribution which is specified for the particular flow motion under investigation, the Reynolds number, and the • Parameters , cr, and ci. • Before we proceed with the discussion of Orr-Sommerfeld equation, we consider the shear stress at the wall that generally can be written as: If the flow is subjected to an adverse pressure gradient, the slope may approach zero and the wall shear stress disappears.

Rayleigh equation • An inviscid flow is defined as the viscous flow with the Reynolds number approaching infinity. • For this special case the Orr-Sommerfeld stability equation reduces to the following Rayleigh equation Rayleigh Equation is a second order linear differential equation and need to satisfy only two boundary conditions:

Solution of Rayleigh Equation • The Rayleigh equation can be readily solved either analytically or numerically. [Rayleigh (1880) ] • Two important theorems on inviscid stability are developed as follows: • Theorem 1 : It is necessary for instability that the velocity profile have a point of inflection. • Theorem 2: The phase velocity cr, of an amplified disturbance must always lie between the minimum and maximum values of U(y). • Rayleigh's result, Theorem 1, led engineers for many year to believe that real (viscous) profiles without a point of inflection such as channel flows and boundary layers with favorable pressure are stable. • It remained for Prandtl (1921) to show that viscosity can indeed be destabilizing for certain wave numbers at finite Reynolds number.

Solution of OS Equations • The Orr-Sommerfeld equation is an eigenvalue problem. • To solve this differential equation, first of all the velocity distribution must be specified. • As an example, the velocity distribution for plane Poisseule flow can be prescribed. • For given Reynolds number and the wavelength, OSE with the boundary conditions provide one eigen function (y) and one complex eigen value c=cr+ici with as the phase velocity of the prescribed disturbance.

Recognition of Stability of Flow • For a given value of disturbances are damped if ci <0 and stable laminar flow persists. • ci > 0 indicates a disturbance amplification leading to instability of the laminar flow. • The neutral stability is characterized by ci = 0. • For a prescribed laminar flow with a given U(y) the results of a stability analysis is presented schematically on a Re Vs Amplitude of disturbance.

Neutral curves of the Orr-Sommerfeld equation

Stability of Blasius BL

Stability map for a plane Poiseulle flow.

Effect of pressure gradient The effect of pressure gradient is equivalent to making velocity profile more stable or unstable. Negative values of pressure gradient signify a favorable pressure gradient. positive values signifyan adverse pressure gradient (pressure increasing downstream),

An intermittently laminar-turbulent flow