Flow potential

In the region ou tside the boundary layer, where the fluid may be assumed to have no viscosity, the mathematical solution takes on the form known as potential flow. This flow is analogous to the flow of heat in a temperature field or to the flow of charge in an electrostatic field. The basic equations of heat conduction (Fourier s law) are 

Here is the flojw of heat per unit time per unit area in the x direction, k is the thermal conductivity, and T is the temperature. An energy balance for some arbitrary region in space (analogous to the procedure shown in finding Eq. 3.32) yields 

Here p is the density and Cy the heat capacity. For constant k this simplifies to 

For steady state, the left term is zero, so that the steady-state heat conduction equation becomes 

This is Laplace s equation, for which solutions are known for many geometries 

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Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

Examples of the theoretieal veloeity distributions in the impeller blades of a eentrifugal eompressor are shown in Figure 6-18. The blades should be designed to eliminate large deeelerations or aeeelerations of flow in the impeller that lead to high losses and separation of the flow. Potential flow solutions prediet the flow well in regions away from the blades where... 

Single gas bubbles in an inviscid liquid have hemispherical leading surfaces and somewhat flattened wakes. Their rise velocity is governed by Bernoulli s theory for potential flow of fluid around the nose of the bubble. This was first solved by G. I. Taylor to give a rise velocity Ug of ... 

Airflow near the hood can be described using the incompressible, irrota-tional flow (i.e., potential flow) model. The potential flow theory is based on... 

Flanged Elliptical Opening The potential flow solution for an elliptical aperture in an infinite wall with a constant potential across the hood face is given by Lamb as... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

Another design method uses capture efficiency. There are fewer models for capture efficiency available and none that have been validated over a wide range of conditions. Conroy and Ellenbecker - developed a semi-empirical capture efficiency for flanged slot hoods and point and area sources of contaminant. The point source model uses potential flow theory to describe the flow field in front of a flanged elliptical opening and an empirical factor to describe the turbulent diffusion of contaminant around streamlines. 

Potential flow theory is used to predict the velocity components (V, V.)... 

Thus it is recommended the simple potential flow model be used to obtain a first estimate for the optimization of the effective capture region in any particular application. Once this has been achieved, the equipment should be built to this specification but with sufficient flexibility to adjust it to obtain the practical optimum effective capture region. 

Potential Flow Model for a Bench Slot Exhaust... 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

A reduced scale of the model requires an increased velocity level in the experiments to obtain the correct Reynolds number if Re < Re for the prob lem considered, but the experiment can be carried out at any velocity if Re > RCj.. The influence of the turbulence level is shown in Fig. 12.40. A velocity u is measured at a location in front of the opening and divided by the exhaust flow rate in order to obtain a normalized velocity. The figure show s that the normalized velocity is constant for Reynolds numbers larger than 10 000, which means that the flow around the measuring point has a fully developed turbulent structure at that velocity level. The flow may be described as a potential flow with a normalized velocity independent of the exhaust flow rate at large distances from the exhaust opening— and far away from surfaces. 

The maximum unsupported tube spans in Table 10-6 do not consider potential flow-induced vibration problems. Refer to Section 6 for vibration criteria. 

The critical part of the valve consists of a synthetic sapphire ball resting on a seat. The seat may be of stainless steel, PTFE or, more usually, also of sapphire. When the flow is directed against the ball the ball moves forward allowing the liquid to flow past it. When the direction of pressure changes resulting in potential flow-back through the valve, the ball falls back on its seat and arrests the flow. 

Figure 6.3.6 further compares and of lean to stoichiometric methane/air mixtures for all five cases— plug flow, potential flow, asymmetric plug flow, radiative plug-flow, and radiative asymmetric plug-flow. The... 

As Wallis (1969) points out, the upper limit of region 4 is with very large bubbles when their rise is dominated by inertial forces. Under these conditions, the terminal rise velocity is readily calculated from potential flow theory and is given by... 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

FlameMaster v3.3 A C+ + Computer Program for OD Combustion and ID Laminar Flame Calculations. FlameMaster was developed by H. Pitsch. The code includes homogeneous reactor or plug flow reactors, steady counter-flow diffusion flames with potential flow or plug flow boundary conditions, freely propagating premixed flames, and the steady and unsteady flamelet equations. More information can be obtained from http //www.stanford.edu/group/pitsch/Downloads.htm. 

The theoretical results are evaluated through comparison with results of numerical integrations for the counterflow configuration with potential-flow boundary conditions. 

Inviscid Flow and Potential Flow Past a Sphere... 

Since ij/ by definition satisfies Eq. (1-9), potential flow solutions can be found by solving Eq. (1-26) for ij/ subject to the required boundary conditions. The pressure field can then be found using Eq. (1-22). 

These results are useful reference conditions for real flows past spherical particles. For example, comparisons are made in Chapter 5 between potential flow and results for flow past a sphere at finite Re. Other potential flow solutions exist for closed bodies, but none has the same importance as that outlined here for the motion of solid and fluid particles. 

This result may be contrasted with potential flow past a sphere, where the streamlines again have fore-and-aft symmetry but p is an even function of 9 so that there is no net pressure force (see Chapter 1). Additional drag components arise from the deviatoric normal stress ... 

Streamline curvature over a very extensive region, and there is infinite drift. On the axis of symmetry, the fluid velocity falls to half the sphere velocity almost two radii from the surface. The corresponding distance for potential flow is 0.7 radii. 

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