Plane stagnation flow

Fig. 12-8 Plane stagnation flow with fluid injection.

An important application of transpiration cooling is that of plane stagnation flow, as illustrated in Fig. 12-8. Solutions for the influence of transpiration on heat transfer in the neighborhood of such a stagnation line have also been worked out in Ref. 3, and the results are shown in Fig. 12-9. As would be expected, gas injection or suction can exert a significant effect on the temperature recovery factor for flow over a flat plate. These effects are indicated in Fig. 12-10, where the recovery factor r is defined in the conventional way as... 

For the heated vertical plate and horizontal cylinder, the flow results from natural convection. The stagnation configuration is a forced flow. In each case the flow is of the boimdai7 Kiyer type. Simple analytical solutions can be obtained when the thickness of the du.st-free space is much smaller than that of the boundary layer. In this case the gas velocity distribution can be approximated by the first term in an expansion in the distance norroal to the surface. Expressions for the thickness of the dust-free space for a heated vertical surface and a plane stagnation flow are derived below. 

It was proposed by the author (Stralmann et al., 1988) that thermophores is could be used to suppress particle deposition on wafers during clean room operations in the microelectronics industry. To estimate the effect of an applied temperature gradient on particle deposition, the flow of filtered air over the surface of a horizontal wafer can be approximated by a stagnation flow (Fig. 3.12), For both the plane and axially symmetric stagnation flows, the gas velocity component normal to the surface and the temperature fields depend only on the distance from the surface. In the absence of natural convection, the gas velocity normal to the surface in the neighborhood of the plane stagnation flow is... 

Figure 3.12 Temperature and velocity distribution in plane stagnation flow showing the thickness of the dust-lree space and the boundary layer. (After Stratmann et al, 1988.)...

As already noted in Section IV,B, turbulence can be described in the plane normal to the main flow by the stagnation flow pattern. For steady laminar stagnation flow, u = aDx and v = — a y far from the solid surface and [2]... 

Deriving the axisymmetric stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.60, and 3.60), but considering flow only in the z-r plane. In general, there may be a circumferential velocity component ui, but there cannot be variations of any variable in the circumferential direction 0. The derivation depends on two principal conjectures. First, the velocity field is presumed to be described in terms of a streamfunction that has the separable form... 

Fig. 6.3 Nondimensional axial and radial velocity profiles for the axisymmetric stagnation flow in the semi-infinite half plane above a solid surface. The flow is approaching the surface axially (i.e., u < 0) and flowing radially outward (i.e., V > 0). The temperature profile, which is the result of solving the thermal-energy equation, is discussed in Section 6.3.6.

Fig. 6.6 Comparison of two alternative stagnation-flow configurations. The upper illustration shows the streamlines that result from a semi-infinite potential flow and the lower illustration shows streamlines that result from a uniform inlet velocity issuing through a manifold that is parallel to the stagnation plane. Both cases are for isothermal air flow at atmospheric pressure and T = 300 K. In both cases the axial inlet velocity is u = —5 cm/s. The separation between the manifold and the substrate is 3 cm. For the outer-potential-flow case, the streamlines are plotted over the same domain, but the flow itself varies in the entire half plane above the stagnation surface. The stagnation plane is illustrated as a 10 cm radius, but the solutions are for an infinite radius.

The flow depicted in Figure 9.13 is the most idealized one representing the stagnation flow of a fluid toward a flat plane (see also Figure 9.14). If we imagine that in Figure 9.14 there is a flat plate at x = 0 and that y = 0 is the line of symmetry, then all the particles that initially are a distance smaller than xq = Ux from the line of symmetry will collide with... 

A number of theoretical (5), (19-23). experimental (24-28) and computational (2), (23), (29-32). studies of premixed flames in a stagnation point flow have appeared recently in the literature. In many of these papers it was found that the Lewis number of the deficient reactant played an important role in the behavior of the flames near extinction. In particular, in the absence of downstream heat loss, it was shown that extinction of strained premixed laminar flames can be accomplished via one of the following two mechanisms. If the Lewis number (the ratio of the thermal diffusivity to the mass diffusivity) of the deficient reactant is greater than a critical value, Lee > 1 then extinction can be achieved by flame stretch alone. In such flames (e.g., rich methane-air and lean propane-air flames) extinction occurs at a finite distance from the plane of symmetry. However, if the Lewis number of the deficient reactant is less than this value (e.g., lean hydrogen-air and lean methane-air flames), then extinction occurs from a combination of flame stretch and incomplete chemical reaction. Based upon these results we anticipate that the Lewis number of hydrogen will play an important role in the extinction process. 

The pitot tube is a device for measuring v(r), the local velocity at a given position in the conduit, as illustrated in Fig. 10-1. The measured velocity is then used in Eq. (10-2) to determine the flow rate. It consists of a differential pressure measuring device (e.g., a manometer, transducer, or DP cell) that measures the pressure difference between two tubes. One tube is attached to a hollow probe that can be positioned at any radial location in the conduit, and the other is attached to the wall of the conduit in the same axial plane as the end of the probe. The local velocity of the streamline that impinges on the end of the probe is v(r). The fluid element that impacts the open end of the probe must come to rest at that point, because there is no flow through the probe or the DP cell this is known as the stagnation point. The Bernoulli equation can be applied to the fluid streamline that impacts the probe tip ... 

Thus, we can replace u00 in Equation (8.36) and apply it to both opposed and wind-aided cases. For upward or wind-aided spread the speed increases as cos (f> increases to the vertical orientation. For downward or opposed flow spread, the speed is not significantly affected by changes in until the horizontal inclination is approached for the bottom orientation (—90 < wind-aided as a stagnation plane flow results from the bottom. Figure 8.19 gives sketches of the... 

Hiemenz (in 1911) first recognized that the relatively simple analysis for the inviscid flow approaching a stagnation plane could be extended to include a viscous boundary layer [429]. An essential feature of the Hiemenz analysis is that the inviscid flow is relatively unaffected by the viscous interactions near the surface. As far as the inviscid flow is concerned, the thin viscous boundary layer changes the apparent position of the surface. Other than that, the inviscid flow is essentially unperturbed. Thus knowledge of the inviscid-flow solution, which is quite simple, provides boundary conditions for the viscous boundary layer. The inviscid and viscous behavior can be knitted together in a way that reduces the Navier-Stokes equations to a system of ordinary differential equations. 

Fig. 6.13 Comparison of streamlines from rotating-disk solutions at two rotation rates. Both cases are for air flow at atmospheric pressure and T = 300 K. The induced inlet velocity is greater for the higher rotation rate. In both cases the streamlines axe separated by 27tA4< = 1.0 x 10-6 kg/s. The solutions are illustrated for a 2 cm interval above the stagnation plane and a 3 cm radius rotation plane. The similarity solutions themselves apply for the semi-infinite half plane above the surface.

Fig. 13.12 Opposed-flow diffusion flame between parallel, axisymmetric, burner faces that are fabricated as honeycomb monoliths. As illustrated, the flame is positioned on the oxidizer side of the stagnation plane. However, depending on the flow conditions as well as the fuel and oxidizer composition, the flame may form on the fuel side of the stagnation plane.

Solve first the nonreacting problem for the cold flow without a flame. Based on the computed velocity profile, determine the position of the stagnation plane. In physical terms, explain the position of the stagnation plane. Estimate an effective strain rate for the cold flow. 

Plot and discuss the structure of the velocity profile, including the position of the new stagnation plane. Where is the flame relative to the stagnation plane Estimate the strain rate for the combusting flow. 

Consider the condition, which determines the velocity of the curved flame front propagation in the channel. Inside the stagnation zone filled by combustion products the pressure is constant and is equal to the value at infinity (when x = oo). Because of Bernoulli s integral along the streamline restricting the stagnation zone, the gas motion velocity remains unchanged. Since at x = oo the flow is plane-parallel (ptJO = const, v — 0), distributions of velocity u and of the stream function are associated with the vorticity distribution ... 

Under the conditions of turbulence, the time-averaged velocity field is symmetric with respect to the free stagnation plane, provided the flow rates from the two nozzles are equal. The mean axial velocity profile has a similar shape to the curve of uju ) vs x. The gradient of the time-averaged axial velocity takes the maximum at the stagnation plane, while it approaches zero near the nozzle. 

Consider two-dimensional air flow normal to a plane surface. If the initial air temperature is 20°C, the surface temperature 80°C. and the air velocity in the free stream ahead of the plate is 1 m/s, plot the variation of heat transfer rate in the vicinity of the stagnation point. 

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