that is, equivalent to the replacement of the function /„(zm^/v) by the Taylor series expansion in powers of orders zu^/v and ignoring all terms but the first.

The combination v/ti* appearing in (3.2.8) is the meaning of the scale of viscous sublayer thickness. The thickness <5V of this sublayer is defined by the expression <5V = avv/u!(!, where av is a numerical factor depending on the selected method of definition of the upper boundary of the viscous sublayer as the level on which the Reynolds stress is only a fraction of the viscous stress. It is usually accepted that av = 5.

At a considerable distance from the wall (at large zujv) the relation between the viscous stress and the Reynolds stress becomes inverse: the former is many times less than the latter. In this case the term v du/dz in (3.2.1) can be neglected. Thus it is assumed that the vertical velocity gradient (but not the velocity itself, whose absolute value depends on the law of its changing in the vicinity of the wall where viscosity becomes apparent) is determined only by u^ and z. Then, from dimensional considerations, du u,

dz z where A is a universal constant.

Integrating (3.2.10) over z we have u = Au^ In z + A1; (3.2.11)

where At is a constant depending on kinematic viscosity.

The layer of liquid where the relation (3.2.11) is fulfilled is called the logarithmic boundary layer. The height 5{ of its lower boundary, as well as <5V, should be a function of u% and v, so that from dimensional considerations <5, = oi^/u^, where a, is one more numerical constant which, according to experimental data, is equal to 30.

There is the so-called buffer sublayer between the viscous, sublayer and the logarithmic layer where the viscous stress and Reynolds stress have the same order of magnitude, and the linear profile of velocity turns smoothly into the logarithmic profile.

Let us present A1 as A1 = u#(A In u^/v + B), where B is a new universal constant, and rewrite Equation (3.2.11) as follows:

In this case the function /„(zt^/v) in (3.2.2) takes the form fu(zu^/v) = A In zujy + B. According to data from measurements in smooth pipes, in rectangular channels with smooth walls, and in the boundary layer over a smooth, flat surface, the values of the constant A (or of the von Karman constant k = 1/a that is uniquely connected to it) and of the constant B turn out to be equal to 2.5 and 5.1 respectively. These values provide a sufficiently precise reproduction of the profile of u/u^ for zujx < 5 and 30 < zuj\ < 500. But in the buffer sublayer located between the viscous sublayer and the logarithmic boundary layer, that is, at 5 < zu^/v < 30 the values of u/u^, calculated from Equations (3.2.8) and (3.2.12), and observed values are slightly different (see Figure 3.1). To avoid unnecessary complications and at the same time to ensure the right asymptotes the domain of definition of the solution (3.2.8) extends to zujv = 11.1, and everywhere after that (for zujv > 11.1) the vertical profile of the mean velocity is described by Equation (3.2.12). In other words, it is suggested that the buffer sublayer has zero thickness, and the thickness of the viscous sublayer is equal to <5V = <), = 1 l.l(v/w5|.).

3.3 Vertical distribution of the mean velocity over an immovable rough surface; roughness parameter; hydrodynamic classification of underlying surfaces

We now consider the vertical distribution of the mean velocity over an immovable rough surface where the average height h0 of undulations is not

Buffer Sublayer
Figure 3.1 Vertical profile of mean velocity in the turbulent flow over a smooth surface. Dots are experimental data from different authors; solid lines are described by Equation (3.2.8) at zuj\ < 5 and by Equation (3.2.12) at zujx > 30.

small compared with the thickness depth scale v/u* of the viscous sublayer. Because elements of roughness have a considerable effect on the distribution of the mean velocity in the vicinity of the underlying surface, then (3.2.2) is replaced by the more general expression

Here the ellipsis means that in (3.3.1) the non-dimensional parameters characterizing a form of roughness elements and their mutual arrangement are omitted.

Let us discuss again the cases of large and small values of z. For small z compared with h0 the above with regard to the dependence of mean velocity on the determining parameters is valid. In this case to use dimensional considerations for revealing general regularities of the vertical distribution of the mean velocity is unpromising. Another problem is the other limiting case when the distance z from the underlying surface is much more than the average height h0 of roughness elements and the thickness scale v/u^ of the viscous sublayer, that is, z » h0 > v/u^. In this case the local properties of the underlying surface do not have any marked effect on the vertical distribution of the mean velocity, the number of determining parameters is reduced, and there is a formula for u (compare with (3.2.11)):

where, however, the constant A1 prescribed by the condition of continuity of velocity at the lower boundary of the domain of definition of solution (3.3.2) depends on the dimensions, form and mutual arrangement of roughness elements controlling the vertical structure of a flow in the near-wall layer and on viscosity.

As before, we represent the constant Ax as A1 = u^(A In u^/v + B) or, what amounts to the same, A1 = u^(A ln(u%h0/v) — A In h0 + B) and then substitute it into (3.3.2). As a result this expression is rewritten as follows:

where B' = u^(A \n(u^h0/v) + B) is the new numerical constant which is a function of the surface Reynolds number Re0 = w^/i0/v and of the form and mutual arrangement of the roughness elements.

We take into consideration that A = 1/k and introduce the definition of the roughness parameter z0 = hQ exp( —kB'). Substitution of this expression in (3.3.3) yields u, z m = — In —, (3.3.4)

k z0

from which it follows that the roughness parameter represents a level where the mean velocity would vanish if the logarithmic law for the profile of the mean velocity were valid up to this level (actually it is valid for z » z0).

For the underlying surface covered by homogeneous densely packed roughness elements, the roughness parameter z0 should depend only on the surface Reynolds number Re0 so that z0 = h0f(Re0), where f(Re0) is a dimensionless universal function. We find asymptotes of this function for small and large values of Re0. Let Re0 « 1 (roughness elements are completely submerged in the viscous sublayer; the effect of the underlying surface on an ambient velocity is caused by the molecular viscosity force). Then the height h0 of the roughness elements has to be excluded from the set of determining parameters. Hence, f(Re0) ~ Re$ \ and the expression for the roughness parameter will take the form where m0 is a numerical factor.

As may be seen in this case, neither the roughness parameter nor, consequently, the profile of the mean velocity depends on the height of the roughness elements. The underlying surface appropriate for this case can be considered as hydrodynamically smooth.

On the other hand, at Re0 » 1 (roughness elements protrude beyond the limits of the viscous sublayer; the effect of the underlying surface on an ambient velocity is caused by the normal pressure force on the roughness elements), kinematic viscosity has to be excluded from the set of determining parameters. This is possible if the function f(Re0) approaches a certain constant value. Designating it as mj we have

The surface complying with this case is termed hydrodynamically rough.

Equations (3.3.5) and (3.3.6) form the basis of the existing classification of underlying surfaces by the surface Reynolds number. According to this classification an underling surface is considered to be hydrodynamically smooth if Re0 < 4, m0 « 0.1; it is considered to be hydrodynamically rough if i?e0 > 60, ml % 0.03; and it is considered to be intermediate (slightly rough) if 4 < Re0 < 60. In the general case where the roughness elements have different forms and are randomly located with reference to each other, the proportionality factors in (3.3.5) and (3.3.6), and the limiting values Re0 for the different types of underlying surface will differ from those mentioned above. In such a situation the hydrodynamic properties of the underlying surface should be described not in terms of the average height of the roughness elements, but, rather, in terms of the so-called equivalent sand roughness defined as the height of irregularities of the underlying surface covered by sand roughness which, for identical u^, complies with the same logarithmic profile of the mean velocity as does that over the surface in question. If we now consider h0 as the height of the equivalent sand roughness, then Equations (3.3.5) and (3.3.6) will be universal in the sense that the numerical factors m0 and ml appearing there will not depend on the form and mutual arrangement of roughness elements.

3.4 Hydrodynamic properties of the sea surface

It is clear from general considerations that in moving away from the sea surface any disturbances created by waves must diminish. Because of this in the range of heights from the upper boundary of the surface atmospheric layer to the upper boundary of the wave sublayer (the sublayer within the limits of which the effect of wave disturbances on the mean wind velocity becomes apparent), the vertical structure of the stationary, horizontally homogeneous, neutrally stratified atmospheric boundary layer over the wave disturbed sea surface should remain the same as over an immovable underlying surface. In short, in the above-mentioned height range the logarithmic law for the profile of the mean velocity must be valid.

Up to the present time many gradient measurements of mean wind velocity in the surface atmospheric layer have been gathered. The vast majority of these measurements concerning comparatively great heights (5-10 m) demonstrate satisfactory fulfilment of the logarithmic law under neutral stratification. This experimental fact provides a possibility of determining, using the data from gradient measurement, the friction velocity Uy. and the roughness parameter z0 of the sea surface connected to the resistance coefficient Cu = u^/uj by the relationship where wa is the mean wind velocity at the conventional measurement height za.

At the same time the experimental data referring to heights commensurate with wave heights demonstrate obvious deviations from the logarithmic distribution, and therefore do not admit their interpretation in terms of the hydrodynamic characteristics of the immovable underlying surfaces. Let us discuss first the experimental data on the logarithmic boundary layer. Formally, nothing prevents intepretation of these in terms of the resistance coefficient Cu or the roughness parameter z0 of the sea surface. But a direct analogy between wind waves and roughness elements of the immovable underlying surfaces turns out to be absolutely inappropriate. A dramatic testimony to this is the cloud of dots on the graph of dependence of the roughness parameter z0 on the average height h0 of waves (Figure 3.2). As can be seen, the roughness parameter of the sea surface has strong variability, amounting to six orders of magnitude (see Kitaigorodskii, 1970). The same feature is typical of the resistance coefficient of the sea surface but by virtue of k z,

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