J. M. Floryan

Stability of Goertler vortices in boundary layers

A formal analysis of Goertler-type instability is presented. The boundary-layer and disturbance equations are formulated in a general, orthogonal, curvilinear system of coordinates constructed from the inviscid flow over a curved surface. Effects of curvature on the boundary-layer flow are analyzed. The basic approximation for the disturbance equations is presented and solved numerically. Previous analyses are discussed and compared with our analysis. It is shown that the general system of coordinates developed in this analysis and the correct order-of-magnitude analysis of the disturbance velocities with two velocity scales leads to a rational foundation for future work in Goertler vortices.

Low Reynolds number flow over cavities

Low Reynolds number flow over rectangular cavities is analyzed. The problem is posed to simulate towing tank experiments of Taneda [J. Phys. Soc. Jpn. 46, 1935 (1979)]. A very good agreement exists between numerical and experimental results. The dividing streamline separates from the cavity side wall below the upper corner. The separation point moves toward this corner when the aspect ratio decreases. Flow structure inside the cavity changes considerably with the aspect ratio. Only corner vortices exist in a cavity of aspect ratio W/h=4.0. A decrease of the aspect ratio leads to the enlargement and eventual merger of these vortices. The merger begins with the formation of a stagnation point separating two vortex centers inside the cavity. These vortex centers become progressively weaker and merge to form a single central vortex in a cavity of aspect ratio W/h=2.0. Further decrease of the aspect ratio results in the enlargement of the new corner vortices and their eventual merger. This process begins in a ...

Instabilities of a liquid film flowing down a slightly inclined plane

Falling films on inclined planes display surface and shear instability modes, the latter being important at small angles β of inclination of the plane. These modes are analyzed with particular attention paid to the effects of surface tension. The results show that the critical Reynolds number of the shear mode is nonmonotonic in either the angle β or the surface‐tension parameter ζ but displays a local minimum at nonzero values of β and ζ. For large Reynolds number, an analysis shows that the shear mode is inviscidly stable, but that the surface mode is unstable.

Stability of wall-bounded shear layers in the presence of simulated distributed surface roughness

Linear stability of wall-bounded shear layers modified by distributed suction has been considered. Wall suction was introduced in order to simulate distributed surface roughness. In all cases studied, i.e. Poiseuille and Couette flows and Blasius boundary layer, wall suction was able to induce a new type of instability characterized by the appearance of streamwise vortices. The effects of an arbitrary suction distribution can, therefore, be assessed by decomposing this distribution into Fourier series and carrying out stability analysis on a mode-by-mode basis, i.e. once and for ever.

Vortex instability in a diverging-converging channel

Linear stability of flow in a diverging–converging channel is considered. The flow may develop under either the fixed mass or the fixed pressure gradient constraint. Both cases are considered. It is shown that under certain conditions the divergence–convergence of the channel leads to the formation of a secondary flow in the form of streamwise vortices. It is argued that the instability is driven by centrifugal effect. The instability has two modes and conditions leading to their onset have been identified. These conditions depend on the amplitude and the length of the channel diverging–converging section and can be expressed in terms of a critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required to create the instability for the specified amplitude of the variations of the channel opening are also given. It is shown that the flow developed under the fixed mass constraint is slightly more unstable than the flow developed under the fixed pressure constraint. This difference increases with an increase of the amplitude of the channel divergence–convergence.

Centrifugal instability of Couette flow over a wavy wall

Linear stability of Couette flow over a wavy wall is considered. It is shown that centrifugal effects may create an instability that leads to the formation of streamwise vortices. The conditions leading to the onset of the instability depend on the amplitude and the wavelength of the waviness and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number and the associated wave number of wall waviness required to create the instability for the specified amplitude of the waviness are also given.

Görtler instability of boundary layers over concave and convex walls

The stability analysis of boundary layers over curved surfaces is presented. A simple inviscid stability criterion is developed and used to identify shear layers which are potentially unstable with respect to the Gortler‐type disturbances. A flow is stable if the velocity magnitude increases with distance away from the wall in the case of convex surfaces and decreases in the case of concave surfaces. The flows with nonmonotonic velocity distributions are therefore always potentially unstable regardless of the type of surface being considered. When viscous stability theory is applied to selected boundary layers the results confirm conclusions based on the inviscid theory. Detailed calculations are carried out for the Blasius boundary layer (monotonic velocity distribution) and a wall jet (nonmonotonic velocity distribution). Results suggest that flows with monotonic velocity distributions become unstable earlier than flows with nonmonotonic velocity distributions.

Two-dimensional instability of flow in a rough channel

Linear stability of a two-dimensional flow in a channel with distributed surface roughness is considered. The structure of the disturbance field is related to the structure of the roughness when the ratio of the respective wave numbers is rational; they are not related if this ratio is irrational. It is shown that the stability problem is not unique in the former case but unique in the latter. It is found that disturbances in the form of traveling waves are destabilized by the presence of the roughness. A very good approximation of the critical Reynolds numbers can be found using only the dominant Fourier mode to represent roughness geometry.

Stability of flow in a wavy channel

Linear stability analysis of flow in a channel bounded by wavy walls is considered. It is shown that wall waviness gives rise to an instability that manifests itself through generation of streamwise vortices. The available results suggest that the critical stability criteria based on the Reynolds number based on the amplitude of the waviness can be formulated.

Spectrally-accurate algorithm for moving boundary problems for the Navier–Stokes equations

A spectral algorithm based on the immersed boundary conditions (IBC) concept is developed for simulations of viscous flows with moving boundaries. The algorithm uses a fixed computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization uses Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. It has been demonstrated that the algorithm delivers the theoretically predicted accuracy in both time and space. Performances of various linear solvers employed in the solution process have been evaluated and a new class of solver that takes advantage of the structure of the coefficient matrix has been proposed. The new solver results in a significant acceleration of computations as well as in a substantial reduction in memory requirements.