L. S. Yao

Natural Convection Along a Vertical Wavy Surface With Uniform Heat Flux

Laminar free convection along a semi-infinite vertical wavy surface has been studied by Yao (1983) for the case of uniform surface temperature. This is a model problem for the investigation of heat transfer from roughened surfaces in order to understand heat transfer enhancement. In many applications of practical importance, however, the surface temperature is nonuniform. In this note, the case of uniform surface heat flux rate, which is often approximated in practical applications and is easier to measure in a laboratory, has been investigated. Numerical results have been obtained for a sinusoidal wavy surface. The results show that the Nusselt number varies periodically along the wavy surface. The wavelength of the Nusselt number variation is half of that of the wavy surface, while the amplitude gradually decreases downstream where the boundary layer grows thick. It is hoped that experimental results will become available in the near future to verify the results of this investigation.

Mixed convection along a wavy surface

The results of a study of mixed-convection flow along a wavy surface are presented. The forced-convection component of the heat transfer contains two harmonics. The amplitude of the first harmonic is proportional to the amplitude of the wavy surface; the second harmonic is proportional to the square of this amplitude. Thus, for a slightly wavy surface, only the influence of the first harmonic can be detected. The natural-convection component is a second harmonic, with a frequency twice that of the wavy surface. Since natural convection has a cumulative effect, the second harmonic eventually becomes the dominant component far downstream from the leading edge where forced convection is the dominant heat transfer mode. The results also demonstrate that the total mixed-convection heat flux along a wavy surface is smaller than that of a flat surface.

Nonlinear instability of travelling waves with a continuous spectrum

Abstract The nonlinear evolution of a continuous spectrum of travelling waves resulting from the growth of unstable disturbances in fully-developed fluid flows is studied. The disturbance is represented in its most general form by a Fourier integral over all possible wavenumbers. The Fourier components of the disturbance quantities are expanded in a series of the linear-stability eigenfunctions, and a set of integro-differential equations for the amplitude density function of a continuous spectrum is derived. No approximations are involved in this reduction; hence, a numerical solution of the integro-differential equations is an exact solution of the Navier-Stokes equations. Numerical integration of the integro-differential equations with different initial conditions shows that the equilibrium state of the flow is not unique after the first bifurcation point, but depends on the waveform of the initial disturbance or, equivalently, on ambient noise which cannot always be controlled in practical situations. Multiple equilibrium states are found to occur at the same value of the dynamic similarity parameters; this implies that any property transported by the fluid can at best be determined within a limit of uncertainty associated with nonuniqueness. A perturbation expansion with multiple time scales is used to show that the equations describing the evolution of monochromatic waves and slowly-varying wavepackets in classical weakly nonlinear theories are special limiting cases of the integro-differential equations near the onset of linear instability. The range of validity of the weakly nonlinear expansions is examined for mixed-convection flow in a heated vertical annulus. The results confirm that weakly nonlinear theories fail to give an adequate description of the physics of the flow even near the onset of linear instability. This is because these theories consider only the most unstable mode, and neglect the contribution from other eigenmodes which can have a large effect on the mean flow distortion. Without considering the leading-order effect of the mean-flow distortion, classical weakly nonlinear instability theories fail to account for proper energy exchanges. The numerical results of the integro-differential equations for the amplitude density function are compared with a direct numerical simulation of the Navier-Stokes equations using a Fourier-Chebyshev spectral method. Complete agreement is found between the two numerical solutions. The solution of the integro-differential equations is simpler than and requires only a small fraction of the computer time necessary for solving the Navier-Stokes equations by a spectral method. The current formulation presents an efficient algorithm to solve the Navier-Stokes equations.

Finite-amplitude instability of mixed convection

The finite-amplitude instability of mixed convection of air in a vertical concentric annulus with each cylinder maintained at a different temperature is studied by use of weakly nonlinear instability theory and by direct numerical simulation. A strictly shear instability and two thermally induced instabilities exist in the parameter space of Reynolds and Grashof numbers. The first thermal instability occurs at low Reynolds numbers as the rate of heating increases, and is called a thermal-shear instability because it is a shear-driven instability induced by thermal effects. The second thermal instability occurs at larger Reynolds number as heating increases, and is also a thermally induced shear instability called the interactive instability. The weakly nonlinear results demonstrate that the thermal-shear instability is supercritical at all wavenumbers. With the shear and interactive instabilities, however, both subcritical and supercritical branches appear on the neutral curves. The validity of the weakly nonlinear calculations are verified by comparison with a direct simulation. The results for subcritical instabilities show that the weakly nonlinear calculations are accurate when the magnitude of the amplification rate is small, but the accuracy deteriorates for large amplification rates. However, the trends predicted by the weakly nonlinear theory agree with those predicted by the direct simulations for a large portion of the parameter space. Analyses of the energy sources for the disturbance show that subcritical instability of the shear and interactive modes occurs at larger wavenumbers because of increased gradient production of disturbance kinetic energy. This is because, at shorter wavelengths, the growth of the wave causes the shape of the fundamental disturbance to change from that predicted by linear instability theory to a shape more favourable for shear-energy production. The results also show that many possibly unstable modes may be present simultaneously. Consequently, all of these modes, as well as all of the possible wave interactions among the modes, must be considered to obtain a complete picture of mixed-convection instability.

Non-Newtonian Natural Convection Along a Vertical Flat Plate With Uniform Surface Temperature

Natural-convection boundary-layer flow of a non-Newtonian fluid along a heated semi-infinite vertical flat plate with uniform surface temperature has been investigated using a four-parameter modified power-law viscosity model. In this model, there are no physically unrealistic limits of zero or infinite viscosity that are encountered in the boundary-layer formulation for two-parameter Ostwald–de Waele power-law fluids. The leading-edge singularity is removed using a coordinate transformation. The boundary-layer equations are solved by an implicit finite-difference marching technique. Numerical results are presented for the case of a shear-thinning fluid. The results indicate that a similarity solution exists locally in a region near the leading edge of the plate, where the shear rate is not large enough to induce non-Newtonian effects; this similarity solution is identical to the similarity solution for a Newtonian fluid. The size of this region depends on the Prandtl number. Downstream of this region, the solution of the boundary-layer equations is nonsimilar. As the shear rate increases beyond a threshold value, the viscosity of the shear-thinning fluid is reduced. This leads to a decrease in the wall shear stress compared with that for a Newtonian fluid. The reduction in the viscosity accelerates the fluid in the region close to the wall, resulting in an increase in the local heat transfer rate compared with the case of a Newtonian fluid.

Taylor-Couette Instability With a Continuous Spectrum

Nonlinear evolution of a continuous spectrum of unstable waves near the first bifurcation point in circular Couette flow has been investigated. The disturbance is represented by a Fourier integral over all possible axial wave numbers, and an integrodifferential equation for the amplitude-density function of a continuous spectrum is derived. The equations describing the evolution of monochromatic waves and slowly varying wave packets of classical weakly nonlinear instability theories are shown to be special limiting cases. Numerical integration of the integrodifferential equation shows that the final equilibrium state depends on the initial disturbance, as observed experimentally, and it is not unique. In all cases, the final equilibrium state consists of a single dominant mode and its harmonics of smaller amplitudes. The predicted range of wave numbers for stable supercritical Taylor vortices is found to be narrower than the span of the neutral curve from linear theory. Taylor-vortex flows with wave numbers outside this range are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability.

Uncertainty of convection

Abstract The nonlinear development of finite amplitude disturbances in mixed convection flow in a heated vertical annulus is studied by direct numerical simulation. The unsteady Navier-Stokes equations are solved numerically by a spectral method for different initial conditions. The results indicate that the equilibrium state of the flow is not unique, but depends on the amplitude and wavenumber of the initial disturbance. In all cases, the equilibrium state consists of a single dominant mode with the wavenumber k f , and its superharmonics. The range of equilibrium wavenumbers k f was found to be narrower than the span of the neutral curve from linear theory. Flows with wavenumbers outside this range, but within the unstable region of linear theory, are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that linearly stable long and short waves can also excite a wave inside this narrow band through nonlinear wave interaction. The results suggest that the selection of the equilibrium wavenumber k f is due to a nonlinear energy transfer process which is sensitive to initial conditions. The consequence of the existence of nonunique equilibrium states is that the Nusselt number cannot always be expressed uniquely as a function of appropriate dimensionless parameters such as the Reynolds, Prandtl and Rayleigh numbers. Any physical quantity transported by the fluid, such as heat and salt, can at best be determined within the limit of uncertainty associated with nonuniqueness. This uncertainty should be taken into account when using any accurately measured values of the heat transfer rate since it is only one of the many possible states for the controllable conditions and geometry. Extrapolating this fact to turbulence, it is our opinion, since the time average will depend on the initial condition, it will not equal to the ensemble average even for stationary turbulence. This is because that the mean flow is not unique for a given Reynolds or Rayleigh number. Consequently, the ergodic hypothesis is not valid. From an application point of view, only the time average has physical significance.

Mixed Convection Along a Semi-Infinite Vertical Flat Plate With Uniform Surface Heat Flux

Mixed-convection boundary-layer flow over a heated semi-infinite vertical flat plate with uniform surface heat flux, placed in a uniform isothermal upward freestream, has been investigated. Near the leading edge, the effect of natural convection can be treated as a small perturbation term. The effects of natural convection are accumulative and increase downstream. In the second region, downstream of the leading-edge region, natural convection eventually becomes as important as forced convection. The boundary-layer equations have been solved by an adaptive finite-difference marching technique. The numerical solution indicates that the series solution of the leading-edge region is included in that of the second region. This property is shared by many developing flows. However, the series solutions of local similarity or local nonsimilarity are only valid for very small distances from the leading edge. Numerical results for the local skin-friction factor, wall temperature, and local Nusselt number are presented for Pr = 1 for a wide range of Gr* x /Re 5/2 x , where Gr* x is a local modified Grashof number and Re x is a local Reynolds number. The results indicate that c fx Re 1/2 x and Nu x Re -1/2 x increase monotonically with distance from the leading edge, where c fx is the local skin-friction factor and Nu x is the local Nusselt number, and approach the free-convection limit at large values of Gr* x /Re 5/2 x , although the velocity distribution differs from the velocity distribution in a free-convection boundary layer.

Natural convection near a small protrusion on a vertical plate

Abstract Natural convection in the vicinity of a small protrusion embedded in the boundary layer on a vertical flat plate is considered. Protrusions of height ~ Le 9 7 and length ~ Le 6 7 , where e = Gr − 1 4 , are analyzed in the context of double-deck theory. The lower deck equations are solved numerically by a hybrid spectral finite difference method. The heat transfer rates are determined for two thermal boundary conditions. In the first case, the protrusion is maintained at the same temperature as the plate, while in the second case, the protrusion is held at a temperature higher than the plate temperature. The effect of boundary layer separation on the heat transfer rate is investigated.

Heat transfer near a small heated protrusion on a plate

Abstract The results of a study of high Reynolds number convective heat transfer from a small heated protrusion on a flat plate are presented. Protrusions of height ~ L Re −5 8 and length ~ L Re −3 8 are analyzed in the context of triple-deck theory. An analytical solution for the local heat transfer rate is presented for protrusions of vanishingly small heights. For protrusions of height ~ 1 on the triple-deck scale, a numerical solution is obtained. The effect of boundary layer separation on the heat transfer rate is investigated. It is found that flow separation leads to high local rates of heat transfer downstream of the separation point.