Theory of a calorimeter at a permeable surface

The calorimetric method for determining the heat flux at a permeable (or sublimating) surface requires the solution of several specific heat-transfer problems, since the calorimeter, bringing about discontinuities in the boundary conditions at the wall (the cessation of blowing, a jump in the temperature of the wall and of its catalytic properties, etc.), introduces perturbations into the boundary layer and measures a heat flux differing from the flux in the absence of a calorimeter. Within the framework of the boundary layer, schematization of such problems is usually based on the isolation of an internal boundary layer (sublayer), which is the region of the effect of the new phenomena at the wall, and develops in the main boundary layer [1–5]. To take account of the effect of the inhomogeneity of the flow in the main boundary layer on heat transfer through the sublayer, here the method of mean-mass values is used, which, as has been demonstrated using various examples in [4] and in the present work, has a good degree of accuracy (even in the neighborhood of the breakaway point) and is suitable for the profiles of an inhomogeneous flow of rather general form. Based on this, for a laminar boundary layer finite formulas are obtained below for the heat flux to a calorimeter of relatively small size at a permeable wall, which can be used for the analysis of experiments.     

Experimental observations of the thermal flow around a square obstruction on a vertical wall in a differentially heated cavity

The transient thermal boundary layer flow around a square obstruction placed at the middle of the hot wall in a differentially heated cavity is visualized using a shadowgraph technique. The results show that the thermal boundary layer flow, which is blocked by the obstruction, firstly forms an intrusion head under the obstruction (the lower intrusion head). Subsequently, the lower intrusion head bypasses the obstruction and reattaches to the down-stream boundary. During the reattachment process, a more complicated flow is induced, and eventually both the lower intrusion head and the thermal boundary layer destabilize. After the lower intrusion head is convected away, the thermal boundary layer flow re-stabilizes. At the quasi-steady state, the thermal boundary layer forms a double-layer structure, which is split into two sections by the obstruction. It is demonstrated that both the transient processes and the quasi-steady state flow structures of the thermal boundary layer are significantly altered by the obstruction in comparison with the case without the obstruction.     

Numerical solution of a free surface problem by a boundary element method

This paper describes a method for the numerical solution of a Riabouchinsky cavity flow. Application of a boundary element method leads to a system of non-linear equations. The mild singularity appearing at the separation point is treated with the introduction of a curved boundary element, which satisfies the exact behaviour of the free boundary in that neighbourhood.     

Thermal-wave heat conduction in a solid cylinder which undergoes a change of boundary temperature

The propagation of thermal waves in a solid cylinder which undergoes a change of its boundary temperature is studied by assuming the validity of Cattaneo-Vernotte's constitutive equation for the heat flux. The hyperbolic energy equation, together with its boundary and initial conditions, is written in a dimensionless form and solved analytically by the Laplace transform method. It is shown that, if the boundary temperature undergoes a step change, the temperature field presents singularities. On the other hand, no singularity is present if the temperature change is achieved by a continuous monotonic evolution of the boundary temperature. However, even in this case, the absolute value of the temperature change in internal points of the cylinder can be greater than that prescribed at the boundary.     

Effect of a space-time source structure simulating a dielectric barrier discharge on the laminar boundary layer

The time-dependent pulse-periodic action of a surface electric discharge on a flat-plate laminar boundary layer is simulated theoretically. The effect of the discharge is estimated within the framework of the numerical solution of the boundary value problem for the time-dependent two-dimensional compressible boundary layer with additional terms in the momentum and energy conservation equations simulating the force and thermal action of the discharge on the gas flow with allowance for the pressure gradient across the boundary layer induced by the corresponding body force component. The effect of certain parameters of the problem formulated above on the gas velocity induced by the discharge in the boundary layer is also estimated.     

Turbulent boundary layer on a catalytic surface in a nonequilibrium dissociating gas

The majority of the studies which consider the flow of a dissociating gas in a turbulent boundary layer are devoted to the investigation of either frozen or equilibrium flows on a flat plate.The frozen turbulent boundary layer has been studied by Dorrance [1], Kutateladze and Leont'ev [2], and Lapin and Sergeev [3]. A study of the effect of catalytic recombination processes at the plate surface on the heat transfer in a frozen turbulent boundary layer was made by Lapin [4].Kosterin and Koshmarov [5], Ginzburg [6], Dorrance [7], and Lapin [8] have studied the turbulent boundary layer on a plate in equilibrium dissociating gas.The calculation of the heat transfer in a turbulent boundary layer on a catalytic plate surface with nonequilibrium dissociation was made by Kulgein [9]. In this study the nonequilibrium nature of the dissociation process was taken into account only in the laminar sublayer, while the flow in the turbulent core was considered frozen. The solution was found numerically using a computer by means of a laborious iteration process.The present paper reports a method for calculating the turbulent boundary layer on a flat catalytic plate with arbitrary dissociation rate. The method, constructed using the assumptions customary for turbulent boundary layer theory, is a successive approximation method. Good convergence of the method is assured by the fact that the effect of the nonequilibrium nature of the dissociation process on the parameter distribution in the boundary layer and, consequently, on the friction and heat transfer may be allowed for merely by finding corrections, usually relatively small, to the distribution of these parameters in the equilibrium or frozen flows. The basis of the study is the two-layer scheme of the turbulent boundary layer. The Prandtl and Schmidt numbers and also their turbulent analogs are taken equal to unity. As the model of the dissociating gas we use the Lighthill model of the ideal dissociating gas [10], extended by Freeman [11] to nonequilibrium flows.     

Stability of a two-phase process involving a planar phase boundary in a thermoelastic solid

This investigation is directed toward understanding the role of coupled mechanical and thermal effects in the linear stability of an isothermal antiplane shear motion which involves a steadily propagatingnormal planar phase boundary in anon-elliptic thermoelastic material. When the relevant process is static — so that the phase boundary does not move prior to the imposition of the disturbance —it is shown to be linearly stable. However, when the process involves a moving phase boundary it may be linearly unstable. Various conditions sufficient to guarantee the linear instability of the process are obtained. These depend on the monotonicity of thekinetic response function — a constitutively supplied entity which relates thedriving traction acting on a phase boundary to the local absolute temperature and the normal velocity of the phase boundary-and, in certain cases, on the spectrum of wave-numbers associated with the perturbation to which the process is subjected. Inertia is found to play an insignificant role in the qualitative features of the aforementioned sufficient conditions. It is shown, in particular, that instability can arise even when the normal velocity of the phase boundary is an increasing function of the driving traction if the temperature dependence in the kinetic response function is of a suitable nature. The instability which is present in this setting occurs only in thelong waves of the Fourier decomposition of the moving phase boundary, implying that the interface prefers to be highly wrinkled.     

Diffraction of SH-waves across a mixed boundary in a plate

The diffraction of SH-waves in an infinite elastic plate is studied under general boundary conditions using the Wiener-Hopf technique. A mixed interface boundary value problem is solved as an illustration.     

Transient flow of an ideal incompressible liquid with a mobile boundary in a rectilinear channel

The behavior of a disturbed liquid-gas boundary is considered in transient axisymmetric flow of an ideal incompressible liquid in a confuser with arbitrary expansion angle. The effect of the confuser aperture angle and the function by which the initial disturbance of the spherical liquid-gas boundary is described on the evolution of the boundary with time is shown.     

Theoretical analysis of laminar film condensation in a rotating cylinder with a scraper

The rimming film condensation on the inside wall of a rotating cylinder with a scraper is analyzed. The whole cylinder is divided into two regions, one is the so-called boundary layer region where the radial velocity of the condensate is much smaller than the peripheric velocity so that the boundary layer theory is assumed to be valid; the other is the scraper region where because of the disturbance of the scraper the boundary layer theory does not apply. The boundary layer integral method in the boundary layer region coupling with the integral momentum theorem across the scraper region provides a method to determine the velocity, temperature, and film thickness distributions, and heat transfer coefficients. An extensive discussion about the previous models is given. The sublayer flow rate constancy principle and the variability principle of the boundary layer thickness (therefore the interface velocity) at the scraper position with respect to the rotational speed are proposed. The present model greatly improved the prediction of the average heat transfer coefficient. Received on 5 January 1998     

Interaction of a turbulent boundary layer with a shock wave at transonic flow velocities

An asymptotic theory of the interaction of a turbulent boundary layer on a plate with a normal shock wave of low intensity has been constructed in various studies [1–4] under the assumption that the averaged velocity of the particles in the boundary layer in front of the interaction region satisfies a logarithmic law. In the present paper a different approach to this problem is proposed based on a power law of the velocity in the undisturbed boundary layer. The obtained results give different estimates for not only the sizes of the characteristic flow regions in the interaction region but also for the shock intensity leading to boundary layer separation.     

Repair of a Plate with a Circular Hole by Applying a Patch

The stressed state of a thin elastic infinite plate with a circular hole covered by a circular patch of a greater radius is considered. The center of the hole coincides with the center of the patch. The patch is attached to the plate along its entire boundary. Stresses are prescribed at infinity on the plate and at the hole boundary. Complex Muskhelishvili potentials are found by the method of power series, and the behavior of stresses on the patch–plate interface and at the hole boundary is studied.     

Approximate dynamic boundary conditions for a thin piezoelectric layer

Approximate dynamic boundary conditions of different orders are derived for the case of a thin piezoelectric coating layer bonded to an elastic material. The approximate boundary conditions are derived using series expansions of the elastic displacements and the electric potential in the thickness coordinate of the layer. All the expansion functions are then eliminated with the aid of the equations of motion and boundary/interface conditions of the layer. This results in boundary conditions on the elastic material that may be truncated to different orders in the thickness of the layer to obtain approximate boundary conditions. The approximate boundary conditions may be used as a replacement for the piezoelectric layer and thus simplify the analysis significantly. Numerical examples show that the approximate boundary conditions give good results for low frequencies and/or thin piezoelectric layers.     

Thermocapillary flow near a “cold corner”

The velocity field in a neighborhood of the point of contact between the free and solid boundaries is studied numerically for the problem of noncrucible zone melting in a two-dimensional model formulation. A distinct Prandtl boundary layer on the solid boundary and a Marangoni boundary layer on the free boundary and high gradients of the longitudinal velocity along the free boundary in the immediate vicinity of the “cold corner” are observed. It is found for the first time that with distance from the solid boundary, the velocity curve has a maximum, which is not typical of the ordinary flow near the solid boundary. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 141–148, May–June, 1998.     

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.     

Flow of a power-law nanofluid past a vertical stretching sheet with a convective boundary condition

A boundary layer flow of a non-Newtonian fluid in the presence of nanoparticles is examined. The flow is caused by a vertical stretching sheet. Convergence of the solution obtained is checked. The values of velocity, temperature, skin friction, and Nusselt number in the boundary layer are obtained.     

Generation of Three-Dimensional Disturbances in a Boundary Layer on Strong Interaction with a Hypersonic Flow

Laminar boundary layer flow over an infinite-span, finite-length flat plate is investigated in the regime of strong interaction with a hypersonic gas flow. Under the assumption that an additional condition dependent on the transverse coordinate can be imposed on the trailing edge of the plate the flow functions are expanded in power series in the vicinity of the leading edge. It is shown that these expansions include an indefinite function dependent on the transverse coordinate. The corresponding boundary value problems are formulated and solved and the eigenvalues are determined. It is established that in this case the two-dimensional boundary layer can rearrange itself into a three-dimensional boundary layer.     

A boundary collocation method for the solution of a flow problem in a complex three-dimensional porous medium

A flow problem in a complex three-dimensional domain with a free surface and mixed-type boundary conditions is solved by the boundary collocation method. The solution is expressed as a combination of source functions distributed all around the domain close to the boundary, plus a special basis function to take care of a corner singularity. The resulting procedure is compared with the boundary integral elements method and is found to be simpler and more flexible to implement and faster to compute.     

Numerical Modeling of the Development of a Streaky Structure in a Boundary Layer at a High Freestream Turbulence Level

The disturbances generated by external turbulence in the boundary layer on a flat plate set suddenly in motion are determined. A turbulent flow calculated by direct numerical simulation is taken as the initial conditions. The solution obtained simulates the initial stage of laminar-turbulent transition in the flat-plate boundary layer at a high turbulence level in the oncoming flow. The solution makes it possible to estimate the effects of different factors, such as nonstationarity, nonlinearity, and the parameters of the freestream velocity fluctuation spectrum, on disturbance enhancement in the boundary layer.