Local Similarity Approximation Research Articles

Variant of the Local Similarity Theory and Approximations of Vertical Profiles of Turbulent Moments of the Atmospheric Convective Boundary Layer

Variant of the Local Similarity Theory and Approximations of Vertical Profiles of Turbulent Moments of the Atmospheric Convective Boundary Layer

Local Non-Similar Solutions for Boundary Layer Flow over a Nonlinear Stretching Surface with Uniform Lateral Mass Flux: Utilization of Third Level of Truncation

The present study aims to examine the effects of uniform lateral mass flux on the boundary layer flow induced by a non-linearly stretching surface. For uniform mass flux, the boundary layer flow does not conform to a similarity solution. The problem may be resolved by the similarity solution only when the transverse velocity at the boundary of the porous stretching surface is of the form vw∼xp−12. In other words, the flow becomes non-similar; to date, this has not been reported in the literature. That is why, in the current study, the local-similarity approximation up to the third level of truncation is utilized to solve the problem. The pseudo-similarity variable, stream function and transformed streamwise coordinate are defined such that the continuity equation is identically satisfied, and the momentum equation reduces to a non-similar dimensionless boundary layer equation. We derived the non-similar equations of the first, second and third levels of truncations and compared the numerical results obtained from different levels of truncations. In order to find numerical solutions to these equations, the built-in MATLAB routine, known as bvp4c, is used. Further, all non-similar terms that appear in the momentum equations are retained without any approximations. The approximations are introduced only in the subsidiary equations and relative boundary conditions. For the case of suction, the rate of increase in the numerical values of skin friction coefficient obtained from the first level of truncation with increasing velocity index parameter is found to be underestimated, while overestimation is found in the case of injection. The numerical results that were obtained from the third level of truncations are plotted against the embedding physical parameters and are then discussed.

Effects of thermal dispersion and stratification on combined convection on a vertical surface embedded in a porous medium

The effects of thermal dispersion and thermal stratification on mixed convection about a vertical surface in a porous medium are studied. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations are solved on the basis of the local similarity approximation. The results indicate that both dispersion and stratification effects have considerable influence on the heat transfer rate.

Variable porosity and thermal dispersion effects on vortex instability of a horizontal natural convection flow in a saturated porous medium

A numerical analysis is made to analyze the variable porosity and thermal dispersion effects on the vortex mode of instability of a horizontal natural convection boundary layer flow in a saturated porous medium. The porosity of the medium is assumed to vary exponentially with distance from the wall. In the base flow, the governing equations are solved by using a suitable variable transformation and employing an implicit finite difference Keller Box method. The stability analysis is based on the linear stability theory and the resulting eigenvalue problem is solved by the local similarity approximations. The results indicate that both effects increase the heat transfer rate. In addition, the thermal dispersion effect stabilizes the flow to the vortex mode of disturbance, while the variable porosity effect destabilizes it.

BOUNDARY AND INERTIA EFFECTS ON VORTEX INSTABILITY OF A HORIZONTAL MIXED CONVECTION FLOW IN A POROUS MEDIUM

Abstract Non-Darcian effects, including the effects of boundary and inertia, are examined on the flow and vortex instability of a horizontal mixed convective boundary layer flow in a high-porosity medium. In the base flow, the governing conservation equations are solved by using a suitable variable transformation and employing an implicit finite-difference scheme. The stability analysis is based on linear stability theory, and the resulting equations are solved on the basis of the local similarity approximations. The results show that the flow becomes more susceptible to the vortex mode of disturbances as buoyancy force is increased. In addition, both the boundary and inertia effects reduce the heat transfer rate and stabilize the flow to the vortex mode of disturbances.

Non-Darcian effects on vortex instability of a horizontal natural convection flow in a porous medium

The non-Darcian effects, which include the inertia, boundary and convective effects, on the flow and vortex instability of a horizontal natural convection boundary flow in a high-porosity medium are examined. In the base flow, the governing conservation equations are solved by using a suitable variable transformation and employing an implicit finite-difference scheme developed by Keller. The stability analysis is based on the linear stability theory, and the resulting equations are solved on the basis of the local similarity approximations. The results indicate that all of these three effects reduce the heat transfer rate. In addition, the combined boundary and convective effects stabilize the flow to the vortex mode of disturbance, while the inertia effect destabilizes it.

Thermal Instability of Forced Convection Boundary Layers

Thermal instability of forced convection boundary layers with nonzero streamwise pressure gradient is examined for moderate to high Prandtl numbers. The analysis is carried out for the family of Falkner-Skan flows, here viewed as the lowest order local similarity approximation of general forced convection boundary layers. Calculated critical Rayleigh numbers and wave numbers are found to be independent of the streamwise pressure gradient in the limiting case of infinite Prandtl number, and only weakly dependent on the streamwise pressure gradient for finite Prandtl number cases when the conduction thickness is employed as the reference length scale.

Mixed Convective Burning of a Fuel Surface With Arbitrary Inclination

An analysis is developed for mixed, forced, and free convective combustion on a flat fuel surface of arbitrary inclination that makes use of the laminar boundary layer approximation to describe the gas flow and of the flame-sheet approximation to describe the gas-phase reactions. A mixed-convection parameter (Rexn + Grxm)1/2n that properly scales the dependent and independent variable fields and a mixed convection ratio (Grxm/Rexn)1/2 that plays the role of the downstream local similarity coordinate are introduced in the nondimensionalization of the equations. It is shown that these two parameters, rather than the standard Reynolds, Rex, and Grashof, Grx, numbers are the optimum choice of governing nondimensional groups for this problem. The values of m and n are selected to obtain a similarity solution of the governing equations in the pure free convection limit for a vertical (m = 2, n = 4) and a horizontal (m = 2, n = 5) surface, which are the cases solved in this work. With this formulation, the solution of the problem provides for both cases smooth transition of all physical variables from one convective limit to the other. Results are obtained from numerical integration of the governing equations and from application of the local similarity approximation. It is shown that the range of validity of local similarity is extended beyond that obtained with alternate formulations and that the proper limits are approached. For use in practical applications, the results suggest that explicit expressions for the mass burning rate and for the fraction of unburnt pyrolyzate can be found that will suffice over the whole range of mixed-flow intensity.

Nonsimilar calculations for free convection diffusion flames

Previously obtained results using a local similarity approximation are compared with results of nonsimilar calculations for the problem of a free convection diffusion flame burning on a vertical fuel surface. Comparisons are presented for burning rates and temperatures.

Burning on a vertical fuel surface with finite chemical reaction rate

The problem of a natural convection, diffusion flame burning on a vertical fuel surface is analyzed. The fuel sample is assumed to be thermally thick and sufficiently small that the flow is laminar. The flame chemistry is approximated by a one-step reaction with Arrhenius kinetics. The analysis also includes the phenomena of gas phase and surface radiation. The boundary layer equations are transformed to a set of ordinary differential equations using a local similarity approximation. Numerical solutions are obtained by a finite difference procedure. Results are presented for a fuel having the properties of PMMA burning in O 2/N 2 mixtures. For this system, it is found that the effects of gas phase radiation are not significant. The surface burning rate is shown to have a strong dependence on surface radiation, but only a weak dependence on chemical reaction rate. Predicted extinction limits are found to depend on both the chemical reaction rate and the surface heat loss rate. The calculated results are compared with experiments using N 2 and CF 3Br, which show the same effect of these additives on burning rate, but give different extinction limits. This qualitative behavior is reproduced by the theoretical results when the “chemical effect” of CF 3Br is modeled by reducing the overall reaction rate.

Effects of radiation on the burning of vertical fuel surfaces

An analysis of laminar, free convective burning of a thermally thick, vertical fuel surface is presented. The analysis differs from previous work on this problem by including the effects of gas phase and surface radiation. Schvab-Zeldovich variables are used and the thin flame assumption is made. The boundary layer equations are transformed to a set of ordinary differential equations by use of a local similarity approximation. Numerical solutions are obtained using a finite difference procedure. Results are presented for PMMA burning in O 2 /N 2 mixtures. It is found that gas phase radiation has a neglible effect on burning rate (for this laminar, vertical configuration). Surface radiation is found to strongly affect the results. Whereas adiabatic solutions exist at all oxygen concentrations in the thin flame formulation (i.e., for infinite chemical reaction rates), the inclusion of surface radiation leads to predictions of extinction at Y O, ∞ =0.12 for complete combustion and up to Y O, ∞ =0.18 for incomplete combustion. At oxygen concentrations above these limits surface radiation increases the convective heat flux and decreases the burning rate relative to the adiabatic results.

Vortex Instability in Buoyancy-Induced Flow over Inclined Heated Surfaces in Porous Media

A linear stability analysis is performed for the study of the onset of vortex instability in free convective flow over an inclined heated surface in a porous medium. The undisturbed state is assumed to be the steady two-dimensional buoyancy-induced boundary layer flow which is characterized by a non-linear temperature profile. By a scaling argument, it is shown that the length scales of disturbances are smaller than those for the undisturbed boundary layer flow, thus, confirming the so-called “bottling effects” whereby the disturbances are confined within the boundary layer. By neglecting the lowest order terms in the three-dimensional disturbances equations, the simplified equations are solved based on the local similarity approximations, wherein the disturbances are assumed to have a weak dependence in the streamwise direction. The resulting eigenvalue problem is solved numerically. The critical parameter and the critical wave number of disturbances at the onset of vortex instability are computed for different prescribed wall temperature distribution of the inclined surface. It is found that the larger the inclination angle with respect to the vertical, the more susceptible is the flow for the vortex mode of disturbances; and in the limit of zero inclination angle (i.e., a vertical heated plate) the flow is stable for this form of disturbances.

Laminar Mixed-Mode, Forced and Free, Diffusion Flames

To explore the intersection of previous studies of inert mixed-mode flows and single-mode diffusion flames, this analysis describes a laminar, forced and free, mixed-mode diffusion flame adjacent to a vertically burning fuel slab. Shvab-Zeldovich variables, boundary-layer and local similarity approximations are used. Five flaming and chemical parameters are found necessary to describe this combusting system. The solution obtained herein includes profiles of velocity, Shvab-Zeldovich variables and thereby enthalpy and species, as well as surface fluxes, flame sheet locations and excess pyrolyzate. For comparison, the finite-difference method is also employed. The local similarity solution for surface gradients lies within 20 percent of the finite-difference results. Good agreement with the aforementioned simpler limiting cases is also obtained. The results indicate that the flow characteristics of a mixed-mode flame are more complicated than a simple superposition of the two limiting flame cases.

Shear flow beneath oceanic plates: Local nonsimilarity boundary layers for olivine rheology

The principle of local similarity, which has been used to model the two‐dimensional boundary layers in the oceanic upper mantle, permits calculation of the temperature, velocity, and stress fields with essentially analytic techniques. Finite difference numerical methods are hard pressed to resolve the detail required by the large variation of viscosity between the lithosphere and the asthenosphere. In this paper the local similarity approximation has been justified by quantitatively evaluating the effect of nonsimilarity due to viscous heating, nonlinear temperature‐ and pressure‐dependent rheology, buoyancy, adiabatic cooling, etc. Nonsimilar effects produce only small modifications of the locally similar boundary layers; important geophysical observables such as surface heat flux and ocean floor topography are given to better than 10% by the locally similar solution. A posteriori evaluations of the terms neglected in the boundary layer simplification of the complete equations have been conducted on the locally similar temperature and velocity profiles close to the spreading ridge. The boundary layer models are valid to depths of 100 km at 3 m.y. and 10 km at 0.3 m.y.

Interaction between a laminar mixing region and an outer flow

The interaction between a laminar mixing region and an outer flow is considered in the hypersonic approximation in terms of pressure. The dividing streamline is found under the condition that the induced pressure is the same on both sides of the mixing region [1]. Analytic solutions are obtained for the displacement thickness, the dividing streamline, and the induced pressure in the asymptotic regions of weak and strong interaction by using the local similarity approximation, which has been justified for mixed flows in [2]. The interaction parameter is determined.

Similarity solution of the boundary-layer equations for laser heatedflows

The laminar boundary-layer equations with local similarity approximation are solved in order to estimate the heat transfer rate to the wall in a laser-heated rocket thruster using pure hydrogen, where the average temperature of the hot plasma core is about 14,000 K and the core Reynolds number based on nozzle throat diameter is about 2000. Under these conditions, the density-viscosity product at the wall can be 10 times the freestream value, and the hydrogen is completely dissociated. Hence the boundary layer equations with variable transport properties are solved by a quasi-linearization technique, and the equilibrium properties of hydrogen are used in the calculations. Velocity profiles are obtained, and the wall shear and wall stagnation enthalpy gradient are plotted against the pressure gradient parameter.

Ignition and flame spreading over a solid fuel:Non-similar theory for a hot oxidizing boundary layer

A more accurate theoretical study of ignition of a solid polymeric fuel in a hot oxidizing flow field is carried out as an extension of the previous work reported in the Thirteenth Combustion Symposium. In the present work, the complete non-similar calculation is carried out, eliminating the previously used local similarity approximation and the integral approximation to solve the condensed-phase energy equation. Comparison of the predictions based on the non-similar calculations indicates that both results predict similar trends of ignition behavior. However, the ignition delay time and the amount of downstream shift in the previous local similarity calculation differ significantly from those in the present non-similar calculation. This comparison has implications for other otheries of boundary layer combustion in the literature that utilize the self-similarity approximation. The flame-spreading behavior is also predicted as an extension of the ignition process. In agreement with the experimental observations reported in the previous work, results of the non-similar calculation predict that, in the case of high freestream oxygen concentration, the flame spreading downstream from the point of first ignition is slow. Reducing the oxygen concentration increases the flame-spreading speed in both directions (upstream and downstream). This aggreement between experiment and theory indicates that, when the flame spreads downstream, convective heat transfer is the dominant process in bringing the local surface element to vigorous pyrolysis, rather than conduction, and when the flame spreads upstream in the low freestream oxygen level case, the surface temperature ahead of the flame front is already high enough to cause spontaneous ignition, and the appearance of flame awaits the evolution of an abundant supply of fuel. Ignition and flame spreading are shown theoretically to depend sensitively on the thermal and transient properties of the diluent gas (e.g., helium, argon, nitrogen).

Strong mass injection at the surface of a blunt body in a supersonic flow

The case of supersonic flow over a blunt body when another gas is injected through the surface of the body in accordance with a given law is theoretically investigated. If molecular transport processes are neglected, the flow between the shock wave and the surface of the body should be regarded as two-layer, that is, as consisting of the flow in the shock layer between the shock wave and the contact surface and the flow in the layer of injected gas. A numerical solution of the problem is obtained near the front of the body and its accuracy is estimated. Approximate analytic solutions are obtained in the injected-gas layer: a constant-density solution and a solution of the boundary-layer type in the local similarity approximation. Near the flow axis the numerical and analytic solutions are fairly close, but at a distance from the axis the assumptions made reduce the accuracy of the approximate solutions. The flow in question can serve as a gas-dynamic model of a series of problems describing the radiant heating of blunt bodies in a hypersonic flow. In the presence of intense radiative heat transfer, vaporization is so great that the thickness of the vapor layer is comparable with the thickness of the shock layer. Moreover, the thermal shielding of various kinds of obstacles in channels through which a radiating plasma flows can be organized by means of the forced injection of a strong absorber. The formulation of a similar problem was reported in [1], but the results of the solution were not given. A two-layer model of the flow of an ideal gas over a blunt body was used in [2, 3] for the analysis of radiative heat transfer. In [2] the neighborhood of the stagnation point is considered. In [3] preliminary results relating to two-layer flow over blunt cones are presented. The solution is obtained by Maslen's approximate method.

Flow in a Magnetic Shock Tube with Unsteady Magnetic Field

Flow in a magnetic annular shock tube is analyzed, in which both the driving and applied magnetic fields vary with time. First, one‐dimensional flow is solved by the hypersonic similarity method for the case in which both fields are proportional to a power of time. Second, the local similarity approximation is described to treat more general case in which the fields are smooth functions of time. Third, a purely numerical integration using Lax's finite difference method is carried out for a flow with a sinusoidal driving magnetic field, in which a typical nonsimilar nature of the solution appears in some parts as well as locally similar ones in other parts of the flow. Next, some experiments on a magnetic annular shock tube using air as the working gas, are described with special interest paid to the effect of unsteady azimuthal magnetic field on the flow, and comparisons with theoretical predictions are reported.

On the diffusion of species in similar boundary layers with continuously varying properties

The solutions derived by Freeman and Simpkins for the diffusion equation in an incompressible chemically frozen boundary-layer flow are extended to include variations in surface and fluid properties by a suitable change of variable. In a worked example for the flat plate case, a comparison is made with the exact series solution due to Inger and the local similarity approximation. For large values of the streamwise co-ordinate the local similarity approximation is shown to be identical to the first-order term of the asymptotic expansions derived herein.