Plane channels

let us consider the flow between two infinite parallel planes at a distance 2h from each other. The coordinate X is measured from one of the planes along the normal. Since the fluid velocity is independent of the coordinate Y, we can rewrite (1.5.1) in the form 

The solution of this equation under the no-slip boundary conditions on the planes (V = 0 for X - 0 and X = 2h) has the form 

Formula (1.5.6) describes the parabolic velocity field in a plane Poiseuille flow symmetric with respect to the midplane X = h of the channel. 

The volume rate of flow per unit width of the channel can be found by integrating (1.5.6) over the cross-section  

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The photograph taken from the paper of Uberoi (1959) and presented in Fig. 1 shows the shape of the flame front propagating in a plane channel. It shows stagnation zones near the channel walls where the gas rests with respect to the flame front. The stagnation zones are caused by the refraction of stream lines at the extreme points of the flame front. 

Consider the flame in a plane channel, and choose the cartesian coordinate system (x,y) moving with flame at a constant velocity equal to... 

The conclusions drawn in previous sections and the method for obtaining the integral relations determining the flame propagation velocity along the plane channel are applicable to the case of the cylindric tubes also. The difference lies in the details of the formulas and in the quantitative results. 

The performed numerical calculations for the case of cylindrical symmetry yielded the same qualitative dependence of the flame surface shape, the propagation velocity, and thickness of the stagnation zone, as iu the case of the plane channel. The quantitative results for the cylindrical tube are somewhat different, e.g., the dimensionless propagation velocity along the tube axis at a real a proves to be 50 percent higher than in the case of plane symmetry. 

Borisov V. I. (1978). On the velocity of uniform flame propagation in a plane channel. FGV14, 26 (in Russian). 

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

Water is heated by passing it through an array of parallel, wide heated plates. The plates thus form a series of parallel plane channels. The distance between the plates is 1 mm and the mean velocity in the channels is 1 m/s. The plates are electrically heated, the two sides of each plate together transferring heat to the water flow at a rate of 1600 W/m2. The water properties can be assumed to be constant and they can be evaluated at a temperature of 50°C. If the flow is assumed to be fully developed, find the rate of increase of mean water temperature with distance along the channels. 

Consider air flow in a plane channel when there is uniform heat flux at one wall and when the other wall is adiabatic. If the inlet air temperature is known, find how the temperature of the heated wall at the exit end of the duct varies with die distance, W, between the two walls. Assume that the mean air velocity is the same in all cases and that the flow is fully developed. 

In a heat exchanger, air essentially flows through a plane channel of length, l. The... 

In order to. illustrate how natural convection in a vertical channel can be analyzed, attention will be given to flow through a wide rectangular channel, i.e., to laminar, two-dimensional flow in a plane channel as shown in Fig. 8.15. This type of flow is a good model of a number of flows of practical importance. 

FULLY DEVELOPED MIXED CONVECTIVE FLOW IN A VERTICAL PLANE CHANNEL... 

Velocity profiles in fully developed mixed convective flow in a vertical plane channel. Results are for assisting flow. 

Air flows vfttically upward at a mean velocity of 1 m/s through a vertical plane channel whose walls are temperatures of 30°C and 40°C, the distance between the walls bding 4 cm. The air enters the channel at a temperature of 15°C. Plot the velocity and temperature distribution in the channel assuming fully developed flow. 

Unidirectional flow through a porous medium can often be approximately modeled as flow through a series of parallel plane channels as show n in Fig. P10.1. Using this model derive an expression for the permeability. K, in terms of the channel size. W, and the porosity, . 

In the general statement of the problem (3.1) and (3.2), a pulsating pressure-driven flow in a plane channel with easily penetrable layers symmetrically placed near both walls can be described by the following equation ... 

The first engineering LES was Deardorff s [27] simulation of plane channel flow. Deardorff used Reynolds (spatial) averaging, applied to a unit cell of the finite difference mesh, to define the larger (or resolved) scales, and introduced the terminology filtered variables. Although only 6720 grid points were used, the comparison with literature laboratory experiments was sufficient favorable for the feasibility of the method to have been established. 

Schumann U (1975) Subgrid Scale Model for Einite Difference Simulations of Turbulent Flows in Plane Channels and Annuli. J Comput Phys 18 376-404... 

Let us introduce the equivalent (or hydraulic ) diameter de by the formula de = 45 /V, where 5 is the area of the tube cross-section and V is the cross-section perimeter. For tubes of circular cross-section, de coincides with the diameter, and for a plane channel, de is twice the height of the channel. 

Formulas (1.6.1) and (1.6.2) have formed the basis of most theoretical investigations on the determination of the average fluid velocity and the drag coefficient in the stabilized region of turbulent flow in a circular tube (and a plane channel of width 2a). The corresponding results obtained on the basis of Prandtl s relation (1.1.21) and von Karman s relation (1.1.22) for the turbulent viscosity can be found in [276,427]. In what follows, major attention will be paid to empirical and semiempirical formulas that approximate numerous experimental data quite well. 

Qualitatively, the picture of stabilized turbulent flow in a plane channel is similar to that in a circular tube. Indeed, in the viscous sublayer adjacent to the channel walls, the velocity distribution increases linearly with the distance from the wall V(Y)/U = y+. In the logarithmic layer, the average velocity profile can be described by the expression [289]... 

In the flow core, the average velocity distribution in a plane channel of width 2h can be approximately described by formulas of the form [212, 289]... 

More detailed information about the structure of turbulent flows in a circular (noncircular) tube and a plane channel, as well as various relations for determining the average velocity profile and the drag coefficient, can be found in the books [138, 198, 268, 276, 289], which contain extensive literature surveys. Systematic data for rough tubes and results of studying fluctuating parameters of turbulent flow can also be found in the cited references. 

Heat and Mass Transfer in a Laminar Flow in a Plane Channel... 

Temperature field. We shall study the heat exchange in laminar flow of a fluid with parabolic velocity profile in a plane channel of width 2h. Let us introduce rectangular coordinates X, Y with the X-axis codirected with the flow and lying at equal distances from the channel walls. We assume that on the walls (at Y = h) the temperature is constant and is equal to Ti for X < 0 and to T2 for X > 0. Since the problem is symmetric with respect to the X-axis, it suffices to consider a half of the flow region, 0 < Y < h. 

Mean flow rate temperature. Nusselt number. The flow rate temperature for a plane channel is given by the formula... 

Now let us consider the case in which a constant heat flux q is given on the walls of a plane channel for X > 0. We assume that for X < 0 the walls are thermally isolated and the temperature tends to a constant T as X - -oo. 

Let us discuss qualitative specific features of convective heat and mass transfer in a turbulent flow through a circular tube and plane channel in the region of stabilized flow. Experimental evidence indicates that several characteristic regions with different temperature profiles can be distinguished. At moderate Prandtl numbers (0.5 < Pr < 2.0), the structure and sizes of these regions are similar to those of the wall layer and the core of the turbulent stream considered in Section 1.6. 

Let us consider the problem of dissipative heating of a fluid in a plane channel of width 2h with isothermal walls on which the same constant temperature is... 

Now let us consider a steady-state hydrodynamically stabilized flow of a non-Newtonian fluid through a plane channel of width 2h. Let us introduce Cartesian coordinates X, with X-axis directed downstream along the lower wall and with coordinate measured inward the channel along the normal to this wall (0 < < 2h). Since the problem is symmetric about the midline = h, it suffices to consider the lower half of the region, 0 < < /i. 

Up to the different notation (APjL -> pg sin a), formula (6.4.15) coincides with the expression (6.2.5) for shear stresses, which was obtained earlier for film flows. Therefore, we can calculate the velocity profile V in a plane channel (in the region 0 < < h), the maximum velocity f/max, and the mean flow rate velocity (V) for nonlinear viscous fluids by formulas (6.2.8)—(6.2.11) and for viscoplastic fluids by formulas (6.2.17)-(6.2.19) if we formally replace pg sin a by AP/L in these formulas. 

For power-law and Shvedov-Bingham fluids, the basic characteristics of flow in a plane channel can be found from Table 6.4, where one must set m = AP/L. 

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