Disturbance velocity

When openings for air supply are used, they should be small, but not too small, since this could generate high, disturbing velocities of incoming air. 

Shibuya (SlO) dealt with the case of the onset of instability in film flow on the outer surface of a vertical tube. By assuming a mixed disturbing velocity of the cosine-hyperbolic cosine type, it was found that the numerical value of the Reynolds number for instability was approximately... 

At the free stream (i.e. as y oo) one requires the disturbance velocity components to decay to zero i.e. 

At the inflow boundary and on top of the computational domain, analytic solution for the disturbance velocity was used in accordance with Eqn. (2.7.1) and (2.7.2). On the flat plate, the no-slip condition simultaneously provides a Dirichlet boundary condition for the stream function and the wall vorticity at every instant of time. 

Figure 2.34 Streamwise disturbance velocity component plotted at a height y = 0.3<5 for the pure convection case (c = U ) at the indicated times, when solution is obtained by solving the Navier-Stokes equation for the problem shown in Fig.2.30...

Figure 4.7 Streamwise disturbance velocity component plotted as a function of x at indicated times for Re = 1000 for ujq = 0.1307 Results are shown for the non-dimensional height of y = 0.278...

For these excitation parameters, existing spatial modes are all damped One set corresponding to the lower frequency (identified as 1 and 2 in Table 4.2) are below the neutral curve and the other set (B) corresponding to the higher frequency (identified as 3, 4 and 5 in the table) are above the neutral curve. In Fig. 4.10, the computed disturbance velocity in streamwise direction, obtained by Bromwich contour integral method is shown for the case of B (wq = 0.15). 

In Fig. 4.12, the computed disturbance velocity in streamwise direction, obtained by Bromwich contour integral method is shown for the case of the... 

To solve the stability problem one is required to solve (6.4.19)- (6.4.38) starting from y = Yoo using the initial condition given in (6.4.39)- (6.4.58). The dispersion relation can be obtained by satisfying the wall boundary condition as given by (6.4.12) in terms of disturbance velocity components and temperature field. In terms of the fundamental solution components, these conditions at the wall can be written as. 

In Fig. 6.11, two sets of eigenfunctions are shown for the case with K = 5 X 10 and Re = 1000. In Fig. 6.11(a) the case corresponds to LVo = 0.1 for which the hydrodynamic mode attains its maximum growth and thus the eigenfunctions once again represent the disturbance velocity components. In Fig. 6.11(b) the case corresponds to loq = 0.7 and the eigenvalue for this case indicates the thermal mode to be at its maximum growth rate. Hence the plotted function corresponds to the disturbance temperature field. 

In Figs. 6.12 to 6.14, similar eigenfunctions are shown for the identical parameter combinations for which the hydrodynamic and thermal modes are most unstable and thus represents disturbance velocity and temperature fields. 

The most important feature of (7-151) is that the disturbance velocities fall off only linearly with distance from the body. Thus, for a body that exerts a net force on the fluid, the disturbance produced is extremely long range in a low-Reynolds-number flow. One important implication is that the motion of such a body will be extremely sensitive to the presence of another body, or other boundaries, even when these are a considerable distance away. For example, the velocity of a sphere moving through a quiescent fluid toward a plane wall under the action of buoyancy is reduced by approximately 12% when the sphere is 10 radii from the wall, and by 35% when it is 4 radii away. 

To proceed, we again formulate the problem in terms of the disturbance velocity and pressure fields (u, p ), namely,... 

In view of the linearity of the Stokes equations, we can construct a solution to this problem as the sum of two solutions, one in which the external flow is the purely extensional part of (8-36) characterized by E, and the other in which the external flow is just the rotational motion associated with the vorticity rotating sphere in the. Here we seek a solution for the extensional component of (8-36). 

The constants c2 and c3 are again arbitrary, apart from the continuity constraint (8-4). To obtain a complete form for the disturbance velocity corresponding to (8-36), we combine (8-39) and (8-40) in the form (8-3), and add the solution for the rotational component of the undisturbed flow ... 

We shall see that the stokeslet solution plays a fundamental role in creeping flow theory. We have already seen in Section E of Chap. 7 that it describes the disturbance velocity far away from a body of any shape that exerts a nonzero force on an unbounded fluid. Indeed, when nondimensionalized and expressed in spherical coordinates, it is identical to the velocity field, (7 151). In the next section we use the stokeslet solution to derive a general integral representation for solutions of the creeping-flow equations. 

In this technique, we seek the disturbance velocity field,... 

Now, neither uf nor uf is exact, because neither satisfies the no-shp condition on the surface of the other particle. For example, the motion of particle A produces a disturbance velocity in the vicinity of B of order Ify a/d and the motion of B similarly produces a disturbance velocity of the fluid in the vicinity of A. Thus, to improve the approximation of the velocity field near particle A, we add to uf a correction uf that is a solution of the creeping-flow equations that vanishes at infinity and satisfies the condition... 

But each of these disturbance velocities is just the superposition of a pair of uniform streaming flows. 

In the analysis outlined earlier in this section, we essentially assumed that the velocities of the particles were fixed, and calculated the resulting hydrodynamic forces. Thus we apply the boundary conditions (8-242) and (8-243) to determine uf and uf, and we see that uf and uf are just the (creeping-flow) disturbance velocities for simple translation of spheres A and B with velocities —uf (A) and -uf (B) through an unbounded fluid. The solutions for uf and uf therefore can be expressed in the same form as (8-247a) and (8-247b), namely,... 

Show that the disturbance velocity field necessary to satisfy no-shp boundary conditions at the surface of the ellipsoid can be expressed in terms of a line distribution of stresset, rotlet, and potential quadrapole singularities of the form... 

The field 11 is called the disturbance velocity field because it represents the difference between the undisturbed field U, which would exist if the body were not present, and the actual velocity in the presence of the body. It is observed that the velocity field u is completely symmetric about y = 0 (all x). Furthermore, when x is large, the major deviations from the free-stream velocity U occur in the vicinity of the x axis (y = 0) that is, u is nonzero only near y = 0 for large x, and this region is called the wake-flow region. 

Finally, to complete the statement of the linear stability problem, we require boundary conditions. Far from the interface, as z oo, we expect the disturbance velocities to decay to zero ... 

We require that the disturbance velocity decay far from the interface (i.e., as z oo). Furthermore, at the fluid interface the normal component of the disturbance velocity satisfies the kinematic condition,... 

Thus the problem of describing the time-dependent evolution of an arbitrary initial disturbance reduces to the solution of a single sixth-order PDE for the disturbance velocity component w. Once w is known, the other disturbance quantities can be calculated. For example, 9 can be obtained from (12-181), and p. u, and v can then be determined by means of (12-180), (12-182), and (12-183). The objective here is to use (12-192) to examine the conditions for growth or decay of an arbitrary initial disturbance in w. Because the governing equation is linear and separable, it will clearly have a solution of the form... 

In addition to the momentum equation, it is also necessary to satisfy the equation of continuity, which for the incompressible fluid considered is V u = 0. The disturbance velocity is also irrotational hence, in addition, V X u = 0 (Lighthill 1978). Therefore the disturbance velocity field is derivable from a gradient of a velocity potential that is, u = V with the potential satisfying Laplace s equation V — 0. This may seem surprising at first, but Laplace s equation can describe a wave motion when boundary conditions are satisfied at a free surface. 

The second step is the linearization of the governing flow equations and boundary conditions assuming the flow to be only slightly disturbed. The momentum equation so linearized is given by Eq. (10.4.3), with the excess or disturbance pressure and u the disturbance velocity. The continuity equation for the disturbance velocity is as before, V (/> = 0. The natural coordinates for the problem are the cylindrical coordinates (r, 6, z) in which Laplace s equation takes the form... 

We assume the neutral disturbance velocities in the dimensionless form as... 

Early stability analyses employed the classical Kelvin-Helmholtz (K-H) theory for two inviscid layers (Kordyban and Ranov [33], Kordyban [34], Wallis and Dobson [35]). However, in referring to gas-liquid flows, pyp C 1, and assuming that the interfacial disturbance velocity equals the (slower) liquid layer velocity, the liquid destabilizing contribution has been degenerated. This results in a rather simple Bernoulli-type transitional criteria, whereby the suction forces in the gas-... 

Greenspan resonance when Ui = Cj, the alongshore component Ui) of the atmospheric disturbance velocity equals the phase speed of the jth mode of edge waves (cj) ... 

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