Energy Integral Equations Research Articles

Unsteady Laminar Boundary Layers Over an Arbitrary Cylinder With Heat Transfer in an Incompressible Flow

Abstract A method is presented for calculating the development of momentum and thermal laminar boundary layers on a heated cylinder of arbitrary shape when the cylinder moves in an incompressible fluid at rest with an unsteady velocity. This analysis is based on solutions to the unsteady momentum and energy-integral equations in conjunction with a set of universal functions, derived from exact solutions to the boundary-layer equations for a specific unsteady problem. These universal functions are given in tabulated form. Those associated with the energy-integral equation are calculated with a Prandtl number of 0.7. The reliability and limitation of these functions are indicated and discussed in the light of several simple problems of which solutions are available. A detailed calculation procedure for the general unsteady problem is given and then followed by a numerical example.

The structure of turbulent shear flow

A theoretical investigation is made of the mixing layer between two streams. The work is divided into four sections. The first involves the solution of the mean problem of laminar and turbulent mixing. The equations of motion are written in terms of a similarity variable. An eddy viscosity hypothesis is made to describe the shear stresses. The similarity equations for both laminar and turbulent problems are solved numerically by an iterative scheme. The second section examines the stability of the mixing layer. The Orr-Sommerfeld equation of hydrodynamic stability is solved numerically. Doth cases of spatial and temporal amplification are examined. The shear layer is shown to be unstable to both spatially and temporally growing disturbances at all Reynolds numbers. A correction is made for the divergence of the mean flow and leads to a value of critical Reynolds number of 12.3. The third section presents a model for the turbulent mixing layer. A set of partial differential equations describing the flow are obtained. A Fourier transform technique is employed to reduce this set to an ordinary differential equation for the fluctuating flow field. The homogeneous form of this equation is solved numerically. The resulting predictions of fluctuating velocity, pressure and their correlations are compared with measured values. The agreement is good in certain cases and this serves as a guide to components of the flow governed by non-linear processes. The final section examines the non-linear growth of the mixing layer through transition from laminar to turbulent flow. A set of integral momentum and energy equations are written in terms of a number of shape parameters of mean and fluctuating flow. The amplitude of disturbances is shown to grow rapidly and reach a limiting value. A sudden growth of the mixing layer thickness through transition is also predicted. Comparison is made with experimental results.

Heat Transfer to an Incompressible Turbulent Boundary Layer and Estimation of Heat-Transfer Coefficients at Supersonic Nozzle Throats

An approximate solution is obtained for the heat transfer to an incompressible turbulent boundary layer with arbitrary freestream-velocity and surface-temperature distributions. The analysis is based on the simultaneous solution of the boundarylayer momentum and energy integral equations and employs several simplifying assumptions, the most important of which is the approximation that the boundary-layer velocity and totaltemperature profiles and the skin-friction coefficient are independent of pressure gradient. Solutions are given for twodimensional and axisymmetric flows. Although the flow in supersonic nozzles is not incompressible, an adaptation of the aforementioned result, termed the quasiincompressible solution, is suggested for calculating the heat transfer to such nozzles. In order to estimate rapidly the maximum rate of heat transfer which occurs near the throat of a supersonic nozzle, a throat approximation is developed which involves only the nozzle supply conditions, throat opening, and throat radius of curvature. The application of these solutions to a sample nozzle gave the following results using the quasi-incompressible analysis: (1) The heat-transfer coefficients based on solution of both the momentum and energy equations were between 0 and 15 per cent lower than the values based on a solution of the momentum equation alone, and (2) the heat-transfer coefficients that included the effects of variable wall temperature were between 0 and 15 per cent lower than the values obtained assuming constant wall temperature. The throat-approximation, heat-transfer coefficients were within 10 per cent of the corresponding quasiincompressible values.

An Approximate Solution of Compressible Turbulent Boundary-Layer Development and Convective Heat Transfer in Convergent-Divergent Nozzles

Abstract A method has been derived for computing the development of the velocity boundary layer, the temperature boundary layer, and the local convective heat transfer in convergent-divergent nozzles. The method is based on approximate solutions of the integral momentum and integral energy equations for compressible turbulent boundary layers. To obtain the solutions, a flow model is adopted for which are prescribed the velocity and temperature profiles in the boundary layer, the skin-friction law, and the relation between heat and momentum transfer. The local heat-transfer coefficient is expressed as an explicit function of the boundary-layer thicknesses. The effects of nozzle size and throat radius of curvature on boundary-layer development and heat transfer are determined for certain similar nozzle contours in common use. The results of the solution are demonstrated by a sample calculation for a conventional rocket nozzle operating under typical conditions. The computed local heat fluxes were found to be in approximate agreement with those measured during rocket-motor tests using a nozzle of the same contour.