Jacek Szumbarski

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.

Transient disturbance growth in a corrugated channel

Transient growth of small disturbances may lead to the initiation of the laminar-turbulent transition process. Such growth in a two-dimensional laminar flow in a channel with a corrugated wall is analysed. The corrugation has a wavy form that is completely characterized by its wavenumber and amplitude. The maximum possible growth and the form of the initial disturbance that leads to such growth have been identified for each form of the corrugation. The form that leads to the largest growth for a given corrugation amplitude, i.e. the optimal corrugation, has been found. It is shown that the corrugation acts as an amplifier for disturbances that are approximately optimal in the smooth channel case but has little effect in the other cases. The interplay between the modal (asymptotic) instability and the transient growth, and the use of the variable corrugation for modulation of the growth are discussed.

Numerical simulation of flows over corrugated walls

Abstract Effectiveness of three methods for simulation of flows over corrugated/rough boundaries has been analyzed. Domain perturbation method, which approximates shape of the boundary, is applicable to corrugation amplitudes up to 10−1 and provides accuracy no better than O(10−1) for such conditions. Its absolute accuracy significantly increases for lower amplitudes especially when higher-order versions of the method are used. Domain transformation method (DTM) enforces flow boundary conditions exactly, provides spectral accuracy but is very laborious in implementation due to a very complex form of the field equations. Direct method (DM) enforces flow boundary conditions with spectral accuracy and is simple in implementation. DM is recommended for most applications. When either corrugation amplitude or its wave number becomes excessively large, DTM is recommended due to its better mode efficiency.

Stability of flow in a channel with vibrating walls

The effects of small wall vibrations in the form of traveling waves on the asymptotic stability of a channel flow are analyzed. It is shown that vibrations may affect flow instability only if they consist of modes with wavelength and frequency close to those of the neutral Tollmien–Schlichting (TS) waves. Vibrations in resonance with the neutral TS waves produce instability that amplifies linearly in time. It is shown that vibrations detuned with the neutral TS waves may lead to an initial transient growth linear in time. It is further demonstrated that this growth may lead to flow destabilization through nonlinear effects, resulting in a significant reduction in the critical Reynolds number. The wealth of possible flow responses for various forms of detuning, including modulated wall vibrations, is also discussed.

Low-Reynolds-Number Instability of the Laminar Flow Between Wavy Walls

Instability of a viscous incompressible flow in a channel with wavy walls is investigated theoretically, numerically and experimentally. Linear stability analysis shows that appropriately chosen wall waviness leads to flow destabilization at surprisingly low Reynolds numbers. The unstable mode of disturbances forms a vortex array, which travels downstream. The remarkable feature is that the most destabilizing waviness does not introduce any additional flow resistance. The outcome of the stability analysis are consistent with the result of direct numerical simulation obtained using CFD finite volume package FLUENT (Ansys Inc.). Preliminary experimental data gained for a channel with appropriately corrugated wall seem to confirm these predictions.

Instability in a channel with grooves parallel to the flow

Flow in a channel with corrugated walls has been studied, with the primary goal of establishing channel geometries that enhance achievable mixing at possibly low drag increase. The wall corrugation has the form of a sinusoidal wave oriented transversely, i.e., the lines of constant elevation (or phase) are parallel to the direction of the flow. The analysis is performed up to the Reynolds numbers resulting in the formation of secondary states. The first part of the analysis is focused on the properties of the two-dimensional, base flow. Mainly, the dependence of the drag on the channel’s geometry is characterized. The second part of the analysis discusses the onset of the three-dimensional traveling wave instability. Linear stability is investigated by the Direct Numerical Simulation of the Navier-Stokes equations. Critical conditions for the onset of instabilities at a range of geometric parameters are determined. Finally, nonlinear saturation of the unstable modes and the resulting secondary flows is ex...

Robust Optimization with Gaussian Process Models

In this chapter, the application of the Gaussian regression models in the robust design and uncertainty quantification is demonstrated. The computationally effective approach based on the Kriging method and relative expected improvement concept is described in detail. The new sampling criterion is proposed which leads to localization of the robust optimum in a limited number of steps. The methodology is employed to the optimal design process of the intake channel of the small turboprop engine.

Flow dynamics in longitudinally grooved duct

Flow in a finite-width rectangular duct with a corrugated top-bottom wall has been studied. The primary goal is to establish geometries that allow early flow destabilization at a possibly low drag increase. The flow is assumed periodic in the streamwise direction and bounded by the duct sidewalls in the spanwise direction; the top and bottom wall corrugations have a form of sinusoidal waves oriented transversely to the flow and form longitudinal grooves; i.e., the lines of constant elevation (or phase) are parallel to the direction of the flow. The analysis is performed up to the Reynolds numbers resulting in the formation of secondary states. The first part of the analysis is focused on the properties of the two-dimensional base flow. Mainly, the dependence of hydraulic losses and drag reducing properties on duct’s geometry is characterized. The second part of the analysis discusses the onset of the three-dimensional travelling wave instability over a wide spectrum of geometric configurations. Linear stability is investigated by means of the direct numerical simulation of the Navier-Stokes equations. Critical conditions for the onset of instabilities at a range of geometric parameters are determined. Finally, the nonlinear saturation of unstable modes and the resulting secondary flows are examined. We have shown that in the state resulting from the nonlinear saturation of the disturbance, the flow becomes more complex while the drag reducing properties of the base flow can be maintained.Flow in a finite-width rectangular duct with a corrugated top-bottom wall has been studied. The primary goal is to establish geometries that allow early flow destabilization at a possibly low drag increase. The flow is assumed periodic in the streamwise direction and bounded by the duct sidewalls in the spanwise direction; the top and bottom wall corrugations have a form of sinusoidal waves oriented transversely to the flow and form longitudinal grooves; i.e., the lines of constant elevation (or phase) are parallel to the direction of the flow. The analysis is performed up to the Reynolds numbers resulting in the formation of secondary states. The first part of the analysis is focused on the properties of the two-dimensional base flow. Mainly, the dependence of hydraulic losses and drag reducing properties on duct’s geometry is characterized. The second part of the analysis discusses the onset of the three-dimensional travelling wave instability over a wide spectrum of geometric configurations. Linear sta...

Unconditionally stable numerical scheme for natural convection problems

Purpose – Both the importance of the natural convection in science and engineering and the shortage of publications in the field of numerical features of time-stepping schemes for the simulation of coupled heat and fluid flow problems motivate the present work. The paper aims to discuss these issues. Design/methodology/approach – The paper presents the unconditionally stable time-stepping scheme for simulation of coupled problems of mass and heat transport. The paper is divided into two parts. The first part concerns the mathematical formulation of the scheme and discusses its implementation. The second part focuses on the numerical simulation and its results. A detailed investigation of the temporal order of the scheme with respect to the L2-norms of the errors of the pressure, velocity, temperature and divergence of velocity fields has also been given. Findings – The work shows that it is possible to formulate a numerical scheme which is unconditionally stable with respect to the time step size. Moreove...

Stability of Poiseuille Flow in a Corrugated Channel

The main objective of the present study is to determine stability properties of channel flow in the presence of distributed surface roughness. Detailed calculations have been carried for two-dimensional roughness in the form of a single Fourier harmonic. The available results demonstrate that such roughness induces a new flow instability that manifests itself through a generation of streamwise vortices. This instability is not active in the case of smooth walls.