Boundary Layer Method Research Articles

Nonlinear Double-diffusive Convection in a Moderate Prandtl Number Fluid

Recent work on nonlinear double-diffusive convection in a low Prandtl number fluid is extended to the case of a moderate Prandtl number G'. The boundary layer method is used by assuming a large Rayleigh number R for different ranges of the diffusivity ratio -c and the solute Rayleigh number R; It is found that the heat and solute fluxes F and F; increase with G', Rand n;', and that F; F- 1 is independent of G'. The flow is shown to have a solute layer of thickness proportional to F-1-2, The horizontal wavenumber which maximizes F is found to be independent of G', but it increases with Rand R;1

Nonlinear Magnetic Convection at Small Prandtl Numbers

Nonlinear magnetic convection is investigated using the modal equations for cellular convection. The boundary layer method is used assuming large Rayleigh number R and small Prandtl number a for different ranges of the Chandrasekhar number Q. The heat flux F is determined for the value of the wave number which maximizes F. For a weak field, F is independent of Q but increases with Ra; for a moderate field, F decreases with Q but increases with R and a; for a strong field, F decreases with Q, increases with R and is ndependent of a. F eventually becomes 0(1) as Q 4- OCR), and the layer becomes stable.

Boundary-layer solutions of single-mode convection equations

Nonlinear thermal convection between two stress-free horizontal boundaries is studied using the modal equations for cellular convection. Assuming a large Rayleigh number R the boundary-layer method is used for different ranges of the Prandtl number σ. The heat flux F is determined for the values of the horizontal wavenumber a which maximizes F. For a large Prandtl number, σ [Gt ] R⅙(log R)−1, inertial terms are insignificant, a is either of order one (for $\sigma \geqslant R^{\frac{2}{3}}$) or proportional to $R^{\frac{1}{3}}\sigma^{-\frac{1}{2}}$ (for $\sigma \ll R^{\frac{2}{3}}$) and F is proportional to $R^{\frac{1}{3}}$. For a moderate Prandtl number, \[ (R^{-1}\log R)^{\frac{1}{9}} \ll \sigma \ll R^{\frac{1}{6}}(\log R)^{-1}, \] inertial terms first become significant in an inertial layer adjacent to the viscous buoyancy-dominated interior, and a and F are proportional to R¼ and \[ R^{\frac{3}{10}}\sigma^{\frac{1}{5}} (\log\sigma R^{\frac{1}{4}})^{\frac{1}{10}}, \] respectively. For a small Prandtl number, $R^{-1} \ll \sigma \ll (R^{-1} \log R)^{\frac{1}{9}}$, inertial terms are significant both in the interior and the boundary layers, and a and F are proportional to ($R \sigma)^{\frac{9}{32}} (\log R\sigma)^{-\frac{1}{32}}$ and ($R \sigma)^{\frac{5}{16}} (\log R \sigma)^{\frac{3}{16}}$, respectively.

On a nonlinear hydromagnetic convection under a rotational constraint

AbstractNonlinear hydromagnetic convection in a horizontal layer of fluid rotating about the vertical axis is investigated using the mean field approximation. The boundary layer method is used assuming large Rayleigh number R for different ranges of the Chandrasekhar number Q and the Taylor number Ta. The heat flux F is determined for the value of the horizontal wave number α which maximizes F. It is shown that, for certain regions of the parameter space (R, Q, Ta), F and α change discontinuously for Ta greater than some critical value (given R and Q). Thus, for Ta about this critical value, wave numbers and heat fluxes of two different values will be predicted simultaneously. Also, for certian regions of the parameter space, the field can facilitate convection, but rotation can facilitate convection only for sufficiently large Ta.

A boundary layer model for mantle convection with surface plates

Summary. A simple, analytical model for mantle convection with mobile surface plates is presented. Our aim is to determine under what conditions free convection can account for the observed plate motions, and to evaluate the thermal structure of the mantle existing under these conditions. Boundary layer methods are used to represent two-dimensional cellular convection at large Rayleigh and infinite Prandtl numbers. The steady-state structure consists of cells with isentropic interiors enclosed by thermal boundary layers. Lithospheric plates are represented as upper surfaces on each cell free to move at a uniform speed. Buoyancy forces are concentrated in narrow rising and decending thermal plumes; torques imparted by these plumes drive both the deformable mantle and overlying plate. Solutions are found for a comprehensive range of cell sizes. We derive an expression for the plate speed as a function of its length, the mantle viscosity and surface heat flux. Using mean values for these parameters, we find that thermal convection extending to 700 km depth can move plates at 1 cm yr-’, while convection through the whole mantle can move plates at 4-5 cm yr-’. Analysis of the steady-state temperature field, for the case of heating from below, shows that the upper thermal boundary layer develops a complex structure, including an ‘asthenosphere’ defined by a local maximum in the geotherm occurring at depths of 50-1 50 km.

Adsorption of brownian hydrosols onto a rotating disc aided by a uniform applied force

The interaction-force boundary layer method is employed to estimate the steady-state rate of deposition under the action of a spatially uniform, applied force field. A simple algebraic equation was derived which predicts values of the rate within ±20% of those computed from a numerical solution of the complete transport equation—including van der Waals attraction, double-layer repulsion, an externally applied force field, fluid motion, diffusion, and the position-dependence of the particle's mobility and diffusion coefficient. This approximation provides a good estimate of the rate whenever the diffusion boundary layer is thick compared to the range of the interaction forces or when aU/ D ∞ ≤ 0.01, where a and D ∞ are the particle's radius and diffusion coefficient, respectively, and U is the normal component of the undisturbed fluid velocity evaluated one particle radius from the disc's surface.

Convective Heat Transfer in MHD Channels and Its Influence on Channel Performance

The limitations of the integral boundary-layer methods and the potential of the differential boundary-layer method in analyzing MHD channel flows are assessed. By showing the sensitivity of results from the integral method to the choice of parameters, the importance of prescribing accurate boundary-layer profiles and wall heat flux in the integral method is emphasized. A mixing-length-type turbulence model for flow on rough walls is developed and validated by comparison with experimental data. The turbulence model is used in a quasi-threedimensional boundary-layer model to evaluate the influence of wall roughness and pressure gradients on the flow characteristics and performance of MHD channels. The behaviors of skin friction and Stanton number calculated from the analytical model are found to differ considerably from the empirical correlations valid for non-MHD flows without pressure gradients. Nomenclature A, B = parameters in mixing length model B = magnetic field Cf = skin friction E = electric field H = total enthalpy IL = current load

A shock tube investigation of heat transfer from dissociated hydrogen to catalytic surfaces

Shock tube experiments are described which compare the heat transfer rate to flat plate nickel and silicon dioxide surfaces from dissociated hydrogen/argon mixtures at temperatures from 2000 to 6000 K. The experimental conditions were chosen in order to permit the effects of surface recombination of hydrogen atoms on the heat-transfer rate to be explored. Comparisons of the experiments with theoretical predictions based on existing laminar boundary layer methods are presented. The results may be interpreted to yield an experimental value tor the catalytic efficiency (γ') of silicon dioxide surfaces to hydrogen atom recombination of 25 × 10 −4.

Determination of the frequencies of the oscillations of a low-viscosity liquid rotating in an axisymmetric vessel

The damping rate and frequencies of the normal modes of a low-viscosity liquid rotating in an axisymmetric vessel are found. The boundary-layer method enables one to express the result in terms of the solutions of the corresponding problem for an ideal liquid.

Calculation of Subsonic Normal Force and Centre of Pressure Position of Bodies of Revolution Using a Slender Body Theory – Boundary Layer Method

SummaryThe aim of this paper is to present a method for predicting the aerodynamic characteristics of slender bodies of revolution at small incidence, under flow conditions such that the boundary layer is turbulent. Firstly a panel method based on slender body theory is developed and used to calculate the surface velocity distribution on the body at zero incidence. Secondly this velocity distribution is used in conjunction with an existing boundary layer estimation method to calculate the growth of boundary layer displacement thickness which is added to the body to produce the effective aerodynamic profile. Finally, recourse is again made to slender body theory to calculate the normal force curve slope and centre of pressure position of the effective aerodynamic profile. Comparisons made between predictions and experiment for a number of slender bodies extending from highly boattailed configurations to ogive-cylinders, and covering a large range of boundary layer growth rates, indicate that the method is useful for missile design purposes.

Reviews and abstracts - The boundary-layer method in diffraction problems

Reviews and abstracts - The boundary-layer method in diffraction problems

Nonlinear Double-diffusive Convection in a Low Prandtl Number Fluid

Nonlinear double-diffusive convection is studied using the modal equations of cellular convection. The boundary layer method is used by assuming a large Rayleigh number R for a fluid of low Prandtl number (J, and different ranges of the diffusivity ratio 7: and. the solute Rayleigh number Rs. The heat and solute fluxes are found to increase with R(J and decrease with Rs. The effect of the solute is stabilizing, although the convection in a fluid with large (J is less affected by the solute concentration. The flow is shown to have a solute layer which thickens as (J, R, .-1 or R;l decreases. It is proved that it is only for this layer that the solute affects the boundary layer structure.

Some Predictions from the Mean Field Equations of Magnetoconvection

Nonlinear magnetic convection is investigated by the mean field approximation. The boundary layer method is used assuming large Rayleigh number R for different ranges of the Chandrasekhar number Q. The heat flux F is determined for wavenumbers CXn which optimize F. It is shown that there are a finite number of modes in the ranges Q ~ R2/3 and R2/3 ~ Q ~ R, and that the number of modes increases with increasing Q in the former range and decreases with increasing Q in the latter range. For Q = 0(R2/3) there are infinitely many modes, and F is proportional to Rl/3 While the optimal F is independent of Q for Q ~ Rl/2, it is found to decrease with increasing Q in the range Rl/2 ~ Q ~ R and eventually to become of 0(1) as Q -> OCR), and the layer becomes stable.

Precritical equilibrium of a thin shallow shell of revolution and its stability

A foundation is given for asymptotic expansions for the axisymmetric precritical equilibrium of a very thin elastic, strictly-convex, shell of revolution with fixed clamping of the edge, and for the derivation of an asymptotic estimate for the critical load for the loss of stability of this equilibrium for axisymmetric shell buckling. The question of the domain of applicability of the formulas for the asymptotic values of the upper critical loads /1,2/ obtained by the boundary layer method is discussed.

Normal oscillations of a rotating liquid drop

The problem of determining the frequencies and profiles of surface waves of a liquid drop rotating under conditions of weightlessness is considered. For an ideal liquid, Ritz's method can be applied to a certain quadratic functional to calculate the oscillation frequencies on a computer. For a low-viscosity liquid, using the boundary-layer method, formulas are obtained for the decay rate and the corrections to the oscillation frequencies. The results of the calculations are presented in the form of graph and a table.

Axial-Flow Compressor Turning Angle and Loss by Inviscid-Viscous Interaction Blade-to-Blade Computation

A method for computation of the flow field around an arbitrary airfoil cascade on an axially symmetric blade-to-blade surface was developed which takes into account the development and separation of the blade surface boundary layers and mixing in the wake. The method predicts the overall fluid turning and total pressure loss in the context of an inviscid-viscous interaction scheme. The inviscid flow solution is obtained from a compressible flow matrix method. The viscous flow is obtained from a differential boundary layer method which calculates laminar, transitional and turbulent boundary layers. Provisions for the calculation of laminar and turbulent separation regions were added to the viscous scheme. The combined inviscid-viscous interaction scheme described yields results which are quantitatively consistent with experimental data. This suggests that the physical basis for the interactive system is correct and justifies continued exploration and use of the method.

Kinetic energy relaxation of a test particle in a dense fluid

The rate constant kε for the relaxation of the kinetic energy of a solute test hard sphere in a dense hard sphere solvent is studied. Microscopic boundary layer and ring kinetic theory methods are used to construct a simple expression for kε. It is found that kε−1 has three additive contributions arising from (a) uncorrelated binary collisions, (b) coupling to the solvent energy density, and (c) coupling to the solvent shear modes. Thus kε−1 is determined by the Enskog collisional relaxation rate and the rates of the hydrodynamic flows in the solvent of heat and momentum. The competition between these effects is studied as a function of particle size and mass ratios and solvent density. The regimes examined at higher density vary from the identical solute–solvent case, where kε is mainly collisional with a small hydrodynamic heat flow correction, to the large, massive solute case, where kε is governed by solvent momentum flow according to a Stokes law type relation.

Prediction of Incompressible Separated Boundary Layers Including Viscous-Inviscid Interaction

A viscous-inviscid interaction calculation procedure is described for viscous flows containing separation bubbles based on an inverse finite-difference boundary layer method and a small disturbance Cauchy integral formulation for the inviscid flow. The viscous-inviscid matching procedure used appears to eliminate the need for smoothing or under-relaxation which is commonly required by other methods. Turbulence modeling for transitional separation bubbles is discussed and two empirically-based transition models are evaluated for bubble flows. Predictions for a laminar and a laminar-turbulent transitional bubble are presented. These predictions are compared with results from other prediction methods and with experimental measurements for the transitional bubble flow.

Three-Dimensional Viscous-Inviscid Coupling Using Surface Transpiration

The development of a method to obtain the subsonic flow field about complex, three-dimensional geometries, using the viscous-inviscid iteration technique to correct the potential flow for viscous effects is described. A panel method for the calculation of three-dimensi onal inviscid flow is used iteratively with a two-dimensiona l, integral boundary-layer method. The important feature is the use of surface transpiration to simulate boundary layer effects. The cross flow is neglected in the boundary layer calculations, which are carried out along chordwise strips on the wing. A doubly underrelaxed iteration scheme has resulted in an extremely stable method, which usually gives satisfactory results after 4-5 iterations. The method has been tested on a wide variety of configurations, and the results have been uniformly satisfactory. It has been demonstrated that the method of surface transpiration is completely equivalent to the displacement thickness approach as a means of representing the boundary layer.

A high-frequency point source of oscillations near a concave mirror

A high-frequency expansion of the problem in question is obtained by the boundary-layer method. This expansion describes the complex phenomenon of a large number of ray reflections from the surface of the mirror.