Ardeshir Hanifi

Transient growth in compressible boundary layer flow

The potential for transient growth in compressible boundary layers is studied. Transient amplification is mathematically associated with a non‐orthogonal eigenvector basis, and can amplify disturbances although the spectrum of the linearized evolution operator is entirely confined to the stable half‐plane. Compressible boundary layer flow shows a large amount of transient growth over a wide range of parameter values. The disturbance size is here measured by a positive definite energy like quantity that has been derived such that pressure‐related transfer terms in its evolution equation mutually cancel. The maximum of the transient growth is found for structures which are independent of the streamwise direction and is found to scale with R2. This suggests that the transient growth originates from the same lift‐up mechanism found to give large growth in incompressible shear flows. The maximum growth is also found to increase with Mach number. In compressible flow, disturbances that experience optimal transi...

On a Stabilization Procedure for the Parabolic Stability Equations

The numerical-stability consequences of the remaining ellipticity in the Parabolic Stability Equations (PSE) are studied. The analysis of Li and Malik of the constant-coefficient Navier-Stokes equations is extended by a detailed analysis of the parabolizing steps. Dropping of the highest streamwise derivative removes the slowest decaying upstream propagating mode, whereas the fastest remains. This mode can be numerically damped, by use of an implicit discretization of the streamwise derivative and a large enough streamwise step size. Suggestions of how to make the equations well-posed by the addition of a term proportional to the truncation error of the implicit scheme are given. This term is easy to implement, does not change the order of approximation and removes the step-size restriction. An explicit formula for the critical step size is also derived, in the modified equations, which shows that the equations are completely stabilized for a properly chosen stabilization parameter.

Adjoint-based optimization of steady suction for disturbance control in incompressible flows

The optimal distribution of steady suction needed to control the growth of single or multiple disturbances in quasi-three-dimensional incompressible boundary layers on a flat plate is investigated. The evolution of disturbances is analysed in the framework of the parabolized stability equations (PSE). A gradient-based optimization procedure is used and the gradients are evaluated using the adjoint of the parabolized stability equations (APSE) and the adjoint of the boundary layer equations (ABLE). The accuracy of the gradient is increased by introducing a stabilization procedure for the PSE. Results show that a suction peak appears in the upstream part of the suction region for optimal control of Tollmien–Schlichting (T–S) waves, steady streamwise streaks in a two-dimensional boundary layer and oblique waves in a quasi-three-dimensional boundary layer subject to an adverse pressure gradient. The mean flow modifications due to suction are shown to have a stabilizing effect similar to that of a favourable pressure gradient. It is also shown that the optimal suction distribution for the disturbance of interest reduces the growth rate of other perturbations. Results for control of a steady cross-flow mode in a three-dimensional boundary layer subject to a favourable pressure gradient show that not even large amounts of suction can completely stabilize the disturbance.

Sensitivity analysis using adjoint parabolized stability equations for compressible flows

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer.

The stabilizing effect of streaks on Tollmien-Schlichting and oblique waves: A parametric study

The stabilizing effect of finite amplitude streaks on the linear growth of unstable perturbations [Tollmien-Schlichting (TS) and oblique waves] is numerically investigated by means of the nonlinear ...

Shape optimization for delay of laminar-turbulent transition

A method using gradient-based optimization is introduced for the design of wing profiles with the aim of natural laminar flow, as well as minimum wave drag. The Euler equations of gasdynamics, the laminar boundary-layer equations for compressible flows on infinite swept wings, and the linear parabolized stability equations (PSE) are solved to analyze the evolution of convectively unstable disturbances. Laminar‐turbulent transition is assumed to be delayed by minimizing a measure of the disturbance kinetic energy of a chosen disturbance, which is computed using the PSE. The shape gradients of the disturbance kinetic energy are computed based on the solutions of the adjoints of the state equations just named. Numerical tests are carried out to optimize the RAE 2822 airfoil with the aim to delay simultaneously the transition, reduce the pressure drag coefficient, and maintain the coefficients of lift and pitch moments. Constraints are also applied on the geometry. Results show a reduction of the total amplification of a large number of disturbances, which is assumed to represent a delay of the transition in the boundary layer. Because delay of the transition implies reduction of the viscous drag, the present method enables shape optimization to perform viscous drag reduction.

The compressible inviscid algebraic instability for streamwise independent disturbances

An inviscid algebraic instability for streamwise independent disturbances in compressible flow is found to be related to Ellingsen and Palm’s [Phys. Fluids 18, 487 (1975)] solution for incompressible flow. For compressible flow, the streamwise disturbance velocity, the density, as well as temperature perturbations grow linearly with time. The effect of viscosity on the inviscid algebraic growth is clarified using a rescaling of the viscous disturbance equations, showing the dependence of the viscous transient growth on the Reynolds number.

Modal Stability TheoryLecture notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013

This article contains a review of modal stability theory. It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spati ...

The attenuation of sound by turbulence in internal flows

The attenuation of sound waves due to interaction with low Mach number turbulent boundary layers in internal flows (channel or pipe flow) is examined. Dynamic equations for the turbulent Reynolds stress on the sound wave are derived, and the analytical solution to the equation provides a frequency dependent eddy viscosity model. This model is used to predict the attenuation of sound propagating in fully developed turbulent pipe flow. The predictions are shown to compare well with the experimental data. The proposed dynamic equation shows that the turbulence behaves like a viscoelastic fluid in the interaction process, and that the ratio of turbulent relaxation time near the wall and the sound wave period is the parameter that controls the characteristics of the attenuation induced by the turbulent flow.

Transition Prediction and Impact on a Three-Dimensional High-Lift-Wing Configuration

The evolution of maximum lift coefficient of a transport aircraft as a function of Reynolds number can be linked to modifications of the laminar-turbulent transition process. In the framework of European project EUROLIFT (I), a task was dedicated to the physical understanding and the numerical modeling of the transition process in high-lift configurations. Then, in the follow-on project EUROLIFT II, a major step is the integration of transition prediction tools within Reynolds-averaged Navier-Stokes (RANS) solvers in order to estimate the impact of transition on performance. This paper presents an overview of the different activities dealing with transition in the EUROLIFT II project.