Influence of turbulence on heat transfer near a stagnation point

A method is developed for calculating the intensification of heat transfer in the neighborhood of a stagnation point (line) of a body in a turbulent (uniform or jet) flow. Conditions for the onset of intensification of heat transfer are found for the first time, together with a universal dimension-less number that determines the intensification coefficient. The results of a calculation agree with the existing experimental data for different classes of flows.     

Laminar heat transfer near an asymmetric stagnation point

This paper considers the laminar boundary layer at an obstacle near the stagnation point of a three-dimensional incompressible potential flow, asymmetric with respect to this point (e.g., for a jet incident at an angle on an obstacle). The effect of compressibility is investigated in the example of a plane subsonic flow. The solution in the close vicinity of the stagnation point is obtained by expanding in series with respect to the longitudinal coordinate, and for the more distant vicinity, the problem is solved by the method of local similarity. It is shown that in this case (in contrast with a symmetric flow [1, 2]) the maximum heat flux does not coincide with the stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–145, November–December, 1976.The author thanks V. V. Lunev for stating the problem and for scientific guidance.     

Heat transfer in a radiating, nonsteady, three-dimensional stagnation flow

A similar solution has been obtained to the problem of simultaneous radiation and convection for nonsteady stagnation point flow over a three-dimensional blunt body with both boundary layer suction and injection. The diffusion approximation is used to characterize the radiative heat flux. The three-dimensional, time-dependent equations of motion and the energy equation have been transformed into a set of ordinary differential equations by the similarity transformation and the resulting ordinary differential equations have been solved numerically. The effects of accelerating and decelerating flow, the three-dimensional geometry, injection and suction, hot and cold wall conditions, and the conduction-to-radiation parameter on the temperature distribution within the flow have been investigated.     

Turbulence enhancement of stagnation point heat transfer on a body of revolution

Enhancement, by free stream turbulence, of convective heat transfer to the stagnation region of a hemispherical-nosed cylinder has been studied. Increases in heat transfer were found to depend primarily on the Reynolds number and turbulence intensity of the free stream, experimental results being most successfully correlated on a NuRe^?0.5 versus TuRe^0.5 basis. Flow visualization studies have demonstrated the validity of a phenomenological model of the enhancement process, predictions of this theory showing good agreement with experimental results. The effect of free stream turbulence on the stagnation point velocity gradient has also been evaluated.     

Entropy generation approach with heat and mass transfer in magnetohydrodynamic stagnation point flow of a tangent hyperbolic nanofluid

Applied Mathematics and Mechanics - This work examines the entropy generation with heat and mass transfer in magnetohydrodynamic (MHD) stagnation point flow across a stretchable surface. The heat...     

Friction and heat transfer in an incompressible fluid near a plane stagnation point

In the neighborhood of a plane stagnation point, the flow and heat transfer of an incompressible fluid are studied. In the inner flow region, the velocity and pressure fields are described by the complete Navier-Stokes equations, and the temperature field is described by the complete energy equation. In the outer flow region, a two-term asymptotic solution of the corresponding equations is obtained. The problem is reduced to the numerical solution of ordinary differential equations. Numerical results are discussed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–65, July–August, 1996.     

Conditions of simulation of stagnation point heat transfer from a high-enthalpy flow

The problem of local simulation of stagnation point heat transfer to a blunt body is solved within the framework of boundary layer theory on the assumption that the simulation subsonic high-enthalpy flow is in equilibrium outside the boundary layer on the model, while the parameters of the natural flow are in equilibrium at the outer edge of the boundary layer on the body. The parameters of the simulating subsonic flow are expressed in terms of the total enthalpyH ₀, the stagnation point pressurep _w and the velocityV ₁ for the natural free-stream flow in the form of universal functions of the dimensionless modeling coefficients=R _m ^* /R _b ^* ( .<1),=V ₁/2H ₀ ( .<1) whereR _m ^* and R _b ^* are the effective radii of the model and the body at their stagnation points. Approximate conditions for modeling the heat transfer from a high-enthalpy (including hypersonic) flow to the stagnation point on a blunt body by means of hyposonic (M1) flows, corresponding to the case ²1, are obtained. The possibilities of complete local simulation of hypersonic nonequilibrium heat transfer to the stagnation point on a blunt body in the hyposonic dissociated air jets of a VGU-2 100-kilowatt induction plasma generator [4, 5] are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 172–180, January–February, 1993.     

Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder

The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analyt- ically using the homotopy analysis method （HAM）. Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numericM and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.     

Prediction of stagnation point heat transfer for a slot jet impinging on a flat surface

The fluid flow and heat transfer for a slot jet impinging on a flat plate has been analysed for different nozzle-to-plate spacing. The available potential flow solution has been used to solve the boundary layer and energy equations by using the Blasius-Frossling series solution method. The friction factor and Nusselt number have been evaluated as a function of the dimensionless distance from the stagnation point. Correlation for the Stanton number at the Stagnation point, is obtained in terms of velocity gradient at the stagnation point and Reynolds number.

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Berechnung des Wärmeübergangs am Staupunkt für einen Strahl, der senkrecht auf eine ebene Fläche trifft

Zusammenfassung Für einen Fluidstrahl, der senkrecht auf eine ebene Platte trifft, wurden für verschiedene Anordnungen von Düse und Platte Strömung und Wärmeübertragung untersucht. Die beschreibende Potentialtheorie wurde verwendet, um die Grenzschicht und Energiegleichungen mit Hilfe der Blasius-Frossling-Reihenentwicklung zu lösen. Reibungsfaktor und Nusseltzahl sind als eine Funktion des dimensionslosen Abstandes vom Staupunkt dargestellt. Die Beziehung für die Stanton-Zahl am Staupunkt ist in den Ausdrücken des Geschwindigkeitsgradienten am Staupunkt und der Reynoldszahl enthalten.

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Nomenclature A ₁ dimensionless coefficient - a dimensionless parameter - b dimensionless parameter - C _f friction factor,C _f= ₀/(1/2w ² ) - C _p specific heat at constant pressure - F ₀ function ofPr and - G ₄ function ofPr and - f ₁ function of - h heat transfer coefficient - k thermal conductivity - l half-width of slot nozzle - Nu Nusselt number,Nu=hl/k - Pr Prandtl number,Pr=v/ - Re Reynolds number,Re=w l/v - St Stanton number,St=Nu/(Re · Pr) - t temperature - t _w wall temperature - t ambient temperature - U dimensionless velocity,U=u/w - U _f dimensionless free-stream velocity,U _f =u _f /w - U _s dimensionless mainstream velocity along the plate,U _s =u _s /w - u velocity component inx-direction - u _f free stream velocity - u _s mainstream velocity along the plate - w velocity component inz-direction - w velocity at the nozzle exit - x coordination along the plate - X dimensionless distance from the stagnation point along the plate,X=x/l - Y ratio ofU _s andU _f ,Y=U _s /U _f - z coordinate perpendicular to the plate - z _n height of the nozzle above the plate - Z dimensionless height of the nozzle above the plate,Z=z _n /l - thermal diffusivity,=k/( C _p) - dimensionless parameter - dimensionless coordinate perpendicular to the plate - viscosity - kinematic viscosity - ₀ shear stress at the wall - stream function     

A turbulence model for the heat transfer near stagnation point of a circular cylinder

A one-equation low-Reynolds number turbulence model has been applied successfully to the flow and heat transfer over a circular cylinder in turbulent cross flow. The turbulence length-scale was found to be equal 3.7y up to a distance 0.05 and then constant equal to 0.185 up to the edge of the boundary layer (wherey is the distance from the surface and is the boundary layer thickness).The model predictions for heat transfer coefficient, skin friction factor, velocity and kinetic energy profiles were in good agreement with the data. The model was applied for Re 250,000 and Tu0.07.Nomenclature µ,C _D Constants in the turbulence kinetic energy equation - C ₁,C ₂ Constants in the turbulence length-scale equation - Skin friction coefficient atx - D Cylinder diameter - F Dimensionless flow streamwise velocityu/u _e - k Turbulence kinetic energy =1/2 the sum of the squared three fluctuating velocities - K Dimensionless turbulence kinetic energyk/u _e ^/2 - I Dimensionless temperature (T–T _w )/(T –T _w ) - l Turbulence length-scale - l _e Turbulence length-scale at outer region - Nu _D Nusselt number - p Pressure - Pr Prandtl number - Pr _t Turbulent Prandtl number - Pr _k Constant in the turbulence kinetic energy equation - R Cylinder radius - Re _D Reynolds number u D/µ - Re _x Reynolds number u x/µ - R _K Reynolds number of turbulence - T Mean temperature - T Mean temperature at ambient - T _s Mean temperature at surface - Tu Cross flow turbulence intensity, - u Mean flow streamwise velocity - u Fluctuating streamwise velocity - u _e Mean flow velocity at far field distance - u Mean flow velocity at ambient - u* Friction velocity - v Mean velocity normal to surface - V Dimensionless mean velocity normal to surface - x,x ₁ Distance along the surface - y Distance normal to surface - Dimensionless pressure gradient parameter - Boundary layer thickness atu=0.9995u _e - Transformed coordinate iny direction - Fluid molecular viscosity - _t Turbulent viscosity - _eff + _t - µ Fluid molecular viscosity at ambient - Kinematic viscosity/ - Density - Density at ambient - _w Wall shear stress - _w,0 Wall shear stress at zero free stream turbulence     

Heat transfer in magnetohydrodynamic flow near a stagnation point

Summary Steady, axisymmetric, magnetohydrodynamic flow with a stagnation point on an infinite plane wall is considered with a magnetic field applied normal to the wall. Solutions are obtained in the form of series for the velocity, magnetic field and temperature when the magnetic field parameter () and the ratio of viscosity to magnetic diffusivity () are small. The case=O(1) is considered briefly when solutions which Meyer³⁾ obtained by physical order-of-magnitude arguments are derived mathematically as expansions in. Some remarks are made on the consistency of extending the results to flow within the boundary layer near the nose of a bluff body.     

Periodic boundary layer near an axisymmetric stagnation point on a circular cylinder

An analysis is presented to investigate the time-mean characteristics of the laminar boundary layer near an axisymmetric stagnation point when the velocity of the oncoming flow relative to the body oscillates. Different solutions are obtained for the small and high values of the reduced frequency parameter. The range of Reynolds number considered was from 0.01 to 100. Numerical solutions for the velocity functions are presented, and the wall values of the velocity gradients are tabulated.     

Prediction of stagnation point heat transfer for a slot jet impinging on a concave semicylindrical surface

A systematic procedure has been laid out for assessment of fluid flow and heat transfer parameters for a slot jet impinging on a concave semicylindrical surface. Based on Walz's modifications of the Karman-Pohlhausen integral method, expressions have been derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. The work then has been extended for the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation has been presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Berechnung des Wärmeübergangs im Staupunkt eines Strahles, der aus einer rechteckigen öffnung auf eine konkave halbzylindrische Fläche auftrifft

Zusammenfassung Es wurde eine systematische Prozedur für die Abschätzung von Strömungs- und Wärmeübergangsparametern für einen Strahl, der auf eine konkave halbzylindrische Fläche auftrifft, aufgestellt. Basierend auf Walz's Modifikationen der Karman-Pohlhausen Integral-Methode, wurden Ausdrücke für die Berechnung der Impulsdicke, der Grenzschichtdicke und die Versetzungsdicke am Staupunkt abgeleitet. Die Arbeit wurde dann auf die Abschätzung der thermischen Grenzschichtdicke und der lokalen Wärmeübertragungskoeffizienten ausgedehnt. Es wird eine Beziehung für die Nusselt-Zahl am Staupunkt als eine Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände von der Austrittsfläche des Schlitzes bis zur Aufprallfläche aufgestellt.

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Nomenclature c _p specific heat at constant pressure - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _.5 Nusselt number based on impinging jet quantities =h _0.50/k - Nu _.5,0 stagnation point Nusselt number =h _{0 0.50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - p(x*) static pressure at nondimensional distancex* from the stagnation point - Re _J jet Reynolds number =U _J W/ - Re _0.5 Reynolds number based on impinging jet quantities =u _{m0 0.50}/ - T temperature - T* nondimensional temperature =(T–T _W)/(T _J–T _W) - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty = _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - W jet nozzle width - x* nondimensional coordinate starting at the stagnation point =x/2 _0.50 - x, y rectangular cartesian coordinates - y coordinate normal to the wall and starting at the wall - ratio of thermal to velocity boundary layer thickness = _T/ _m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _.50 jet half width at the plane of impingement if the plate were not there - d_.5 free jet (half width) thickness whereu=u _m/2 - _m inner boundary layer thickness atu =u _m - _T thermal boundary layer thickness - nondimensional coordinate normal to wall =y/ _m - _T nondimensional coordinate normal to wall =y/ _T - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity = / - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

Temperature-dependent heat sources or sinks in a stagnation point flow

Summary An analytical study has been made to determine the heat transfer characteristics of a stagnation point flow in which there are temperature-dependent heat sources or sinks. Results have been obtained for both strong and weak sources or sinks for a Prandtl number of 0.7. An analytical method, applicable to all Prandtl numbers, was utilized which circumvented the need for extensive numerical solutions and which, at the same time, provided a closed-form representation for the heat transfer. A few numerical solutions were carried out to verify the method.Nomenclature a _i constants depending on Prandtl number - c _p specific heat at constant pressure - f dimensionless velocity variable - g function defined by equation (13) - g _n functions of (n=1, 2, 3,...) - k thermal conductivity - Pr Prandtl number, c _p /k - q heat transfer rate per unit area at surface - Q heat flux parameter, q/k(u ₁/)^1/2 - S rate of heat generation or removal per unit volume (divided by c _p ) - T static temperature; T _w , wall temperature; T , free-stream temperature - u ₁ proportionality constant for free-stream velocity - U free-stream velocity - v normal velocity component - x coordinate measuring distance along surface from stagnation point - y coordinate measuring distance normal to surface - heat generation parameter, equation (3) - dimensionless normal coordinate, - dimensionless temperature - _n functions of (n=1, 2, 3,...) - absolute viscosity - kinematic viscosity - density     

Prediction of stagnation point heat transfer for a single round jet impinging on a concave hemispherical surface

This paper deals with a systematic procedure for assessment of fluid flow and heat transfer parameters for a single round jet impinging on a concave hemispherical surface. Based on Scholkemeier's modifications of the Karman-Pohlhausen integral method, expressions are derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. This is followed by the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation is presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Bestimmung des Staupunktes bei der Wärmeübertragung für einen einzelnen Strahl, der auf eine konkave halbkugelige Oberfläche trifft

Zusammenfassung Diese Arbeit beschäftigt sich mit dem systematischen Verfahren der Bewertung von Fluidströmungen und Wärmeübertragungsparametern für einen einzelnen runden Strahl, der auf eine konkave halbkugelförmige Oberfläche trifft. Das Verfahren beruht auf Scholkemeiers Modifikation des Karman-Pohlhausen Integrationsverfahrens. Ausdrücke sind für die Berechnung der Impuls-Dicke, der Grenzschichtdicke und der Verschiebungsdicke am Staupunkt hergeleitet worden. Dies ist aus der Berechnung der thermischen Grenzschichtdicke und des lokalen Wärmeübertragungskoeffizienten abgeleitet worden. Es wird eine Gleichung für die Nusselt-Zahl am Staupunkt als Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände vom Strahlaustrittspunkt bis zum Auftreffpunkt auf die Oberfläche vorgestellt.

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Nomenclature c _p specific heat at constant pressure - d diameter of single round nozzle - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _0.5 Nusselt number based on impinging jet quantities=h _0.50/k - Nu _{0.5, 0} stagnation point Nusselt number=h _{0 0,50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - R radius of curvature of the hemisphere - Re _J jet Reynolds number=U _Jd/ - Re _0.5 Reynolds number based on impinging jet quantities=u _{m0 0.50}/ - T temperature - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions (Fig. 1) - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty= _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - x* non-dimensional coordinate starting at the stagnation point=x/2 _0.50 - x, y rectangular Cartesian coordinates - y coordinate normal to the wall starting at the wall - ratio of thermal to velocity boundary layer thickness= _T/_m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _0.50 jet half width at the plane of impingement if the plate were not there - _m inner boundary layer thickness atu=u _m - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity=/ - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point