The Stability of Laminar Flows - 2 P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Laminar flows have a fatal weakness …

Flow Quality in A Wind Tunnel Forced Draught Fan

Construction of A Wind Tunnel

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 .

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.

An intermittently laminar-turbulent flow

Intermittency Factor

On Set of Turbulence

Definition A Fluid motion in which velocity, pressure, and other flow quantities fluctuate irregularly in time and space. “Turbulent Fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be observed.” - Hinze “Turbulence is due to the formation of point or line vortice on which some component of the velocity becomes infinite.:” -Jean Leray

What is turbulence? Unsteady, aperiodic motion in which all three velocity components fluctuate, mixing matter, momentum, and energy. Time

First Methods on Analyzing Turbulent Flow Reynolds (1895) decomposed the velocity field into a time average motion and a turbulent fluctuation - Likewise f stands for any scalar: u, v, w, T, p, where: Time averaged Scalar

Averaging Navier Stokes equations Substitute into Steady incompressible Navier Stokes equations Instantaneous velocity fluctuation around average velocity Average velocity time Continuity equation:

Averaging of Continuity Equations

Time Averaging Operations