The disappearance of viscous and laminar wakes in complex flows

The singular effects of steady large-scale external strain on the viscous wake generated by a rigid body and the overall flow field are analysed. In an accelerating flow strained at a positive rate, the vorticity field is annihilated owing to positive and negative vorticity either side of the wake centreline diffusing into one another and the volume flux in the wake decreases with downwind distance. Since the wake disappears, the far-field flow changes from monopolar to dipolar. In this case, the force on the body is no longer proportional to the strength of the monopole, but is proportional to the strength of the far field dipole. These results are extended to the case of strained turbulent wakes and this is verified against experimental wind tunnel measurements of Keffer (1965) and Elliott & Townsend (1981) for positive and negative strains. The analysis demonstrates why the total force acting on a body may be estimated by adding the viscous drag and inviscid force due to the irrotational straining field. Applying the analysis to the wake region of a rigid body or a bubble shows that the wake volume flux decreases even in uniform flows owing to the local straining flow in the near-wake region. While the wake volume flux decreases by a small amount for the flow over streamline and bluff bodies, for the case of a clean bubble the decrease is so large as to render Betz's (1925) drag formula invalid. To show how these results may be applied to complex flows, the effects of a sequence of positive and negative strains on the wake are considered. The average wake width is much larger than in the absence of a strain field and this leads to diffusion of vorticity between wakes and the cancellation of vorticity. The latter mechanism leads to a net reduction in the volume flux deficit downstream which explains why in calculations of the flow through groups of moving or stationary bodies the wakes of upstream bodies may be ignored even though their drag and lift forces have a significant effect on the overall flow field

The disappearance of laminar and turbulent wakes in complex flows

The singular effects of steady large-scale external strain on the viscous wake generated by a rigid body and the overall flow field are analysed. In an accelerating flow strained at a positive rate, the vorticity field is annihilated owing to positive and negative vorticity either side of the wake centreline diffusing into one another and the volume flux in the wake decreases with downwind distance. Since the wake disappears, the far-field flow changes from monopolar to dipolar. In this case, the force on the body is no longer proportional to the strength of the monopole, but is proportional to the strength of the far field dipole. These results are extended to the case of strained turbulent wakes and this is verified against experimental wind tunnel measurements of Keffer (1965) and Elliott & Townsend (1981) for positive and negative strains. The analysis demonstrates why the total force acting on a body may be estimated by adding the viscous drag and inviscid force due to the irrotational straining field.Applying the analysis to the wake region of a rigid body or a bubble shows that the wake volume flux decreases even in uniform flows owing to the local straining flow in the near-wake region. While the wake volume flux decreases by a small amount for the flow over streamline and bluff bodies, for the case of a clean bubble the decrease is so large as to render Betz's (1925) drag formula invalid.To show how these results may be applied to complex flows, the effects of a sequence of positive and negative strains on the wake are considered. The average wake width is much larger than in the absence of a strain field and this leads to diffusion of vorticity between wakes and the cancellation of vorticity. The latter mechanism leads to a net reduction in the volume flux deficit downstream which explains why in calculations of the flow through groups of moving or stationary bodies the wakes of upstream bodies may be ignored even though their drag and lift forces have a significant effect on the overall flow field

Aeroacoustic and aerodynamic applications of the theory of nonequilibrium thermodynamics

Recent developments in the field of nonequilibrium thermodynamics associated with viscous flows are examined and related to developments to the understanding of specific phenomena in aerodynamics and aeroacoustics. A key element of the nonequilibrium theory is the principle of minimum entropy production rate for steady dissipative processes near equilibrium, and variational calculus is used to apply this principle to several examples of viscous flow. A review of nonequilibrium thermodynamics and its role in fluid motion are presented. Several formulations are presented of the local entropy production rate and the local energy dissipation rate, two quantities that are of central importance to the theory. These expressions and the principle of minimum entropy production rate for steady viscous flows are used to identify parallel-wall channel flow and irrotational flow as having minimally dissipative velocity distributions. Features of irrotational, steady, viscous flow near an airfoil, such as the effect of trailing-edge radius on circulation, are also found to be compatible with the minimum principle. Finally, the minimum principle is used to interpret the stability of infinitesimal and finite amplitude disturbances in an initially laminar, parallel shear flow, with results that are consistent with experiment and linearized hydrodynamic stability theory. These results suggest that a thermodynamic approach may be useful in unifying the understanding of many diverse phenomena in aerodynamics and aeroacoustics

A steady separated viscous corner flow

An example is presented of a separated flow in an unbounded domain in which, as the Reynolds number becomes large, the separated region remains of size 0(1) and tends to a non-trivial Prandtl-Batchelor flow. The multigrid method is used to obtain rapid convergence to the solution of the discretized Navier-Stokes equations at Reynolds numbers of up to 5000. Extremely fine grids and tests of an integral property of the flow ensure accuracy. The flow exhibits the separation of a boundary layer with ensuing formation of a downstream eddy and reattachment of a free shear layer. The asymptotic (’triple deck’) theory of laminar separation from a leading edge, due to Sychev (1979), is clarified and compared to the numerical solutions. Much better qualitative agreement is obtained than has been reported previously. Together with a plausible choice of two free parameters, the data can be extrapolated to infinite Reynolds number, giving quantitative agreement with triple-deck theory with errors of 20% or less. The development of a region of constant vorticity is observed in the downstream eddy, and the global infinite-Reynolds-number limit is a Prandtl-Batchelor flow; however, when the plate is stationary, the occurrence of secondary separation suggests that the limiting flow contains an infinite sequence of eddies behind the separation point. Secondary separation can be averted by driving the plate, and in this case the limit is a single-vortex Prandtl-Batchelor flow of the type found by Moore, Saffman & Tanveer (1988); detailed, encouraging comparisons are made to the vortex-sheet strength and position. Altering the boundary condition on the plate gives viscous eddies that approximate different members of the family of inviscid solutions

Stability and Vortex Shedding of Bluff Body Arrays

The primary purpose of this study was to develop an understanding of the stability of laminar flow through bluff body arrays, and investigate the nature of the unsteady vortex shedding regime that follows. The flow was numerically investigated using a specially developed multi-domain spectral element solver. Important criteria in the solver development were flexibility, efficiency, and accuracy. Flexibility was critical to the functionality of the code, as arrays of varying geometry were investigated. Efficiency with a high degree of accuracy was also of primary importance, with the code implemented to run efficiently on today's massively parallel architectures. Numerical two-dimensional stability analysis of the flow in several configurations of inline and staggered array geometries was performed. The growth rate, eigenfunction, and frequency of the disturbances were determined. The critical Reynolds number for flow transition in each case was identified and compared to that of flow over a single body. Based on the solutions of the laminar flow, a one-dimensional analytical analysis was performed on selected velocity profiles in the wake region. The results of this analysis were used to guide the interpretation of the two dimensional results and formulate a general theory of stability of inline and staggered bluff body arrays. The nature of the flow in the unsteady regime following the onset of instability was examined for an inline and a staggered arrangement. Particular attention was focused on the vortex shedding which was visualized and quantified through computation of the flow swirl, a quantity which identifies regions of rotary motion. The conditions required for the generation of leading edge vortex shedding were identified and discussed. Finally, a third geometry related to the inline and staggered arrays was considered. Flow solution data for this geometry is presented and its suitability as a model for louvered arrays was discussed.Air Conditioning and Refrigeration Project 11

Chemical reaction on heat and mass transfer flow past a vertical cylinder embedded in non-Darcy porous medium

An unsteady two dimensional free convection flow of a viscous incompressible fluid past an impulsively started semi-infinite vertical cylinder adjacent to a non-Darcian porous media in the presence of chemical reaction of first order is investigated. The governing boundary layer equations are formulated with appropriate boundary conditions and are solved using an implicit finite-difference method of Crank-Nicholson type. The problem is shown to be controlled by seven thermophysical and hydrodynamic dimensionless parameters, namely thermal Grashof number (Gr), species Grashof number (Gm), Darcy number (Da i.e. permeability parameter), Forchheimer number (Fs i.e. second order inertial porous parameter), Prandtl number (Pr), Schmidt number (Sc) and chemical reaction parameter (K1).The effects of thermophysical parameter on the transient dimensionless velocity, temperature and concentration are illustrated graphically. Also, the effects of the various thermo-physical parameters on the Skin friction, Nusselt number and Sherwood number are presented and discussed. This model finds applications in polymer production, manufacturing of ceramics or glassware and food processing

The onset of instability in unsteady boundary-layer separation

The process of unsteady two-dimensional boundary-layer separation at high Reynolds number is considered. Solutions of the unsteady non-interactive boundary-layer equations are known to develop a generic separation singularity in regions where the pressure gradient is prescribed and adverse. As the boundary layer starts to separate from the surface, however, the external pressure distribution is altered through viscous-inviscid interaction just prior to the formation of the separation singularity; hitherto this has been referred to as the first interactive stage. A numerical solution of this stage is obtained here in Lagrangian coordinates. The solution is shown to exhibit a high-frequency inviscid instability resulting in an immediate finite-time breakdown of this stage. The presence of the instability is confirmed through a linear stability analysis. The implications for the theoretical description of unsteady boundary-layer separation are discussed, and it is suggested that the onset of interaction may occur much sooner than previously thought

Jet mixing under the influence of a pressure gradient

Theoretical analysis of jet mixing under influence of non-constant pressure gradien