Global Stability Analysis of Shear Flows

This chapter introduces the flow instabilities and their role in the flow transition from laminar to turbulent. How the knowledge of instabilities can help extend the flow’s laminar regime by delaying transition and thus reducing drag, fuel consumption, noise and CO\(_2\) emission. The transition process and different stages of flow transitions are explained for the boundary layer flows. The methods of investigation are also explored in brief.

This chapter introduces the global stability approach and the need for such an approach to study the stability of non-parallel flows. We first introduce the two commonly used approximations which result in a problem solvable locally at one streamwise location. These are the parallel flow assumption and the weakly non-parallel flow assumption. Some applications of each are described. Then we introduce strongly non-parallel flows and the global stability analysis with a few examples. The problem formulation and the numerical discretization schemes used in the present work are discussed next. This is followed by the method of solving the problem numerically, the techniques used and the validation.

This chapter presents the global stability characteristics of the divergent channel. The Jeffery–Hamel (JH) and straight-divergent-straight (SDS) channel flows are considered to study the effect of divergence on flow stability. Both flows’ characteristics are compared, and sensitivity to the sub-critical Reynolds number is studied. The sensitivity of the spectra has been studied for different gird sizes, sponging strength and length, and Neumann and extrapolated type boundary conditions. A detailed analysis of the temporal and spatial structure of the small disturbances is reported for the diverging channel flow.

This chapter presents the effect of convergence and divergence of a channel simultaneously on flow stability. The large value of wall surface waviness has been considered to study its effect on the critical Reynolds number. The temporal and spatial structure of the small disturbances is analysed for the forward, reverse and asymmetric geometry of the convergent-divergent channel. The three mechanisms of the instabilities, exponential instability, ratchet growth in space and localized large transient growth in time, are discussed.

This chapter presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at the inflow boundary, fully non-parallel and non-similar. Linearized Navier-Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations and homogeneous boundary conditions form a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in the azimuthal direction. The Chebyshev spectral collocation method and Arnoldi’s iterative algorithm are used for the numerical solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wavenumbers. The largest imaginary part of the computed eigenmodes is negative; hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while moving downstream. The global modes of the axisymmetric boundary layer are more stable than that of the 2D flat-plate boundary layer at low Reynolds number. However, at a higher Reynolds number, they approach to 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at a low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.

This paper presents the linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cone. The base flow is considered parallel to the axis of the cone at the inlet. The angle of attack is zero, and hence, the base flow is axisymmetric. The favourable pressure gradient develops in the streamwise direction due to the cone angle. The Reynolds number is calculated based on the cone radius (a) at the inlet and free-stream velocity (\(U_{\infty }\)). The base flow velocity profile is fully non-parallel and non-similar. Therefore, linearized Navier-Stokes equations (LNS) are derived for the disturbance flow quantities in the spherical coordinates. The LNS is discretized using Chebyshev spectral collocation method. The discretized LNS and the homogeneous boundary conditions form a general eigenvalues problem. Arnoldi’s iterative algorithm is used for the numerical solution of the general eigenvalues problem. The Global temporal modes are computed for the range of Reynolds number from 174 to 1046, semi-cone angles \(2^\circ \), \(4^\circ \), \(6^\circ \) and azimuthal wavenumbers from 0 to 5. It is found that the Global modes are more stable at a higher semi-cone angle \(\alpha \) due to the development of a favourable pressure gradient. The effect of transverse curvature is reduced at higher semi-cone angles (\(\alpha \)). The spatial structure of the eigenmodes shows that the flow is convectively unstable. The spatial growth rate (\(A_x\)) increases with the increase in semi-cone angle (\(\alpha \)) from \(2^\circ \) to \(6^\circ \). Thus, the effect of an increase in semi-cone angle (\(\alpha \)) is to reduce the temporal growth rate (\(\omega _i\)) and to increase the spatial growth rate (\(A_x\)) of the Global modes at a given Reynolds number.

This chapter presents a global stability analysis of the two-dimensional incompressible boundary layer with the effect of streamwise pressure gradients. A symmetric wedge flow with different non-dimensional pressure gradient parameters (\(\beta _{H}\)) has been considered. The pressure gradient (\({\text {d}}p/{\text {d}}x\)) in the flow direction is zero for \(\beta _{H} = 0\), favourable (negative) for \(\beta _H > 0\) and adverse (positive) for \(\beta _H < 0\). The base flow is computed by the numerical solution of the Falkner-Skan equation. The displacement thickness (\(\delta ^*\)) at the inflow boundary is considered for computing the Reynolds number. The governing stability equations for perturbed flow quantities are derived in the body-fitted coordinates. The stability equations are discretized using Chebyshev spectral collocation method. The discretized equations and boundary conditions form a general eigenvalues problem and are solved using Arnoldi’s algorithm. The global temporal modes have been computed for \(\beta _H=0.022\), 0.044 and 0.066 for favourable and adverse pressure gradients. The temporal growth rate (\(\omega _i\)) is negative for all the global modes. The \(\omega _i\) is smaller for the favourable pressure gradient (FPG) than that of the adverse pressure gradient (APG) at the same Reynolds number (\({\text {Re}} = 340\)). Thus, FPG has a stabilization effect on the boundary layer. Comparing the spatial eigenmodes and spatial amplification rate for FPG and APG show that FPG has a stabilization effect while APG has a destabilization effect on the disturbances.

This chapter discusses local and global stability analysis of the wall jet. The boundary layer and shear flow phenomena make the wall jet a special case. An inflexion point in the base flow velocity profile and viscous effect near the wall makes it dual stability characteristics. The Reynolds number is computed based on the maximum velocity \(U_{\max }\) and wall jet thickness (\(\delta \)). The global stability computations are done for different inlet Reynolds numbers of 30, 40, 50, 80, 100 and 200 for different domain lengths.