Internal Forces in a Viscous Fluid

Hydrodynamic Equations of the Sea

3.9 Internal Forces in a Viscous Fluid

Hydrodynamic Equations of the Sea 91

imately - ( d p / d x ) dx d y dz, with the minus sign indicating that the force is directed toward decreasing x (i.e., toward lower pressure). Similar argu- ments apply to the other faces, so that the vector sum of force per unit vol- ume over the entire element is

-(: ^{? +}

^aP

^ay

^{j ^ +} _(3.51)

where, as before, and

k

are unit vectors in the x, y , and z directions, respectively. Hence the pressure forces on a fluid are given by the negative gradient of the pressure, which acts in the down-gradient direction much like a potential function.

Viscous Forces

Next we derive the term in the momentum equation describing viscous forces, which generally act to dissipate and disperse quantities and to damp- en or otherwise modify instabilities. If p is the molecular viscosity of water (with a numerical value of 1.075 x ^{10 ~} kg (m s)

-'

or 0.01075 poise at T

= 20"C), then the form of the term due t o molecular viscosity will be shown to be p

v

*u. Now the damping due to molecular viscosity is very small and its direct effect on larger-scale motions of the sea is essentially negligible;

however, on very small scales of motion (e.g., capillary waves, acoustic waves, and small-scale turbulence), its effects become appreciable. There exist un- resolved theoretical problems in large-scale turbulence in both oceans and atmospheres, in that most motions are far too complicated and detailed to be described on all scales at once, The first-order effect of instabilities and turbulence on the larger motions is to provide a sink, i.e., a mechanism for dissipation and diffusion of large-scale momentum and energy. These ef- fects are often modeled by the inclusion of an artificially large anisotropic eddy viscosity in the momentum equation, whose numerical value depends on the scales in question, but whose functional form is analogous to that of molecular viscosity. Similarly, eddy diffusivities for heat and salinity will be used in the governing equations that will be introduced later in order to model those processes on larger scales. Such terms are said to parameterize the very complicated processes occurring on scales smaller than the ones of interest, and they have proven t o be both useful and moderately effective, if treated with care. We include these terms in the equations that will follow, briefly justifying their use by invoking Reynolds stresses and mixing length theory in Section 3.12 and referring the reader to the references for addi- tional information.

Helmholtz instability evolving into a turbulent layer and finally diffusing out to reduce the gradient. [From Thorpe, S. A., J. fluid Mech. (1971).]

Hydrodynamic Equations of the Sea 93 Figure 3.14 is intended to provide an intuitive understanding of the ef- fects of turbulent eddy diffusion. These photographs show how a series of small-scale nonlinear waves on a fluid density interface are driven to a tur- bulent state by flow instability, which then acts to diffuse the interface and weaken the vertical density and velocity gradients. Thus diffusion of larger- scale density and momentum occurs during such exchanges, but with numer- ical values of the eddy diffusivity that are orders of magnitude larger than the molecular value. It is clear that this process is very complicated in detail and that a more simplified description of the effect on larger-scale dynamics is required-unless, of course, the object of interest is the instability process itself.

A u ( z = z , )

-

^{+u(z = 2 2 )}

- -

^{+u(z = z 3 )}

Fig. 3.15

Lx

Schematic of changes in fluid velocity field in the horizontal plane, at various depths. The fluid exhibits both shear and divergence.

To derive the term for viscous body forces, consider a fluid flow having the general properties shown in Fig. 3.15, with a velocity u(x,t) (as indicated, by arrows) varying in all three dimensions in both strength and direction.

The flow field shown exhibits both velocity divergence (or dilitation) and shear in both the horizontal (x,y) and vertical (z) directions, where the diver- gence is proportional to a u j / d x j , and the shear t o duj/dxj (with i # 1). Thus there are nine partial derivatives, duj/axj (with i,j = x,y,z) describing the spatial variations in current velocity. These nine derivatives are proportion- al to the components of .the stress tensor, ^T~ (of dimensions N m-’), describing viscous, expansion, and pressure forces acting on a unit area wi- thin the body of the fluid. For the viscous forces, the proportionality fac- tors are the coefficients of viscosity, Ai. Here we are allowing for the fact that, in some fluids, the effective viscosity may be anisotropic (i-e., it may vary with direction). Such is the case in the ocean and atmosphere, although for those fluids, the pure molecular viscosity, p , is isotropic. We will assume that A ; includes the scalar molecular viscosity, p , in addition t o the turbu- lent viscosity. T o derive the relationship between rjj and duj/axj, consider the x component of current, u, to be stressed from above (by wind, for ex- ample) and thus t o be varying in the vertical (Fig. 3.16). The shear stress is transmitted downward by viscous and pseudoviscous forces that have the net result of transferring horizontal momentum in the vertical direction. For small distances, dz, it is found experimentally that the stress is proportional

Fig. 3.16 Schematic of vertically sheared flow showing changes of horizontal ve- locity and of internal shear stress with depth.

to the velocity difference existing between z

+

^{(dz/2) and z}

-

( dz /2) ; in- versely to the distance, dz; and to increase directly with the area, dx dy. Upon expansion of the speed and stress elements in Taylor series, the net stress component, ru, can be written as rvr = A,(au/az) (see Fig. 3.16).

However, in addition to the vertical variation of horizontal velocity, there is another contribution to rn from the horizontal variation of vertical ve- locity; this term transmits a net stress either if the fluid is compressible or if the viscosity varies in space (see Eq. 3.57 ahead); this is required so that a fluid that is in uniform rotation at angular velocity ^a, and whose velocity is thus u = o x r, is subject to no viscous body forces. We therefore add a term given by A,aw/az to the stress element to obtain'

au aw

az ax

ru = A , - + A , -

.

From the symmetry of this element, it is clear that ru = r,,.

(3.52)

'Here we have discussed only the stress due t o velocity variations in the presence of viscous shear forces. In fact, the total stress tensor for a fluid is much more com- plicated than described to this point, and has terms due to the scalar pressure, p , the velocity divergence, V ' u , as well as the shear, and is given by

Thus the pressure and the velocity divergence are diagonal, as represented by the Kronecker delta, 6 v , while the shear contributes off-diagonal terms as well. We have derived p as a distinct force, as given by Eq. 3.51, but it is part of the stress tensor nevertheless, The quantity p"

+

^{% p} is called the volume viscosity, p " ; its correct representation is not altogether clear for fluids. It is ordinarily assumed that pu = 0, so that p " = - % p . Then the viscous forces may be represented entirely by p

and its generalization for eddy motions, A. See the more advanced texts in the Bib- liography for a discussion of this arcane subject.

Hydrodynamic Equations of the Sea 95

Note the indices for 7 : the first index,

z,

denotes the fact that the velocity change occurs over the coordinate

z ,

and that the stress is exerted over an area perpendicular to

z ,

i.e., an element in the x,y plane. The second index denotes the current component and hence stress direction in question. The viscosity coefficient A, indicates that the stress due to this element is trans- mitted in the

z

direction, Similar reasoning can be applied to the remaining term and indeed, all eight elements of the stress tensor.

Consider again our small rectangular parallelepiped containing the parcel of fluid immersed in the flow of Fig. 3.15. At time t = t o , the nine stress- es, T ! ~ , are the forces per unit area exerted over each face in the directions indicated in Fig. 3.17. At a small time, A t , later, the stresses have deformed the cube somewhat, although by assumption it still retains its original mass, d 3 m = p d x d y dz. If the fluid is also incompressible, p is a constant and hence the volume, d 3 x = d x d y dz, must also be constant, if slightly dis- torted geometrically.

By noting that the nine elements of stress shown in Fig. 3.17 are function- ally related to the velocity components in a form similar to Eq. 3.52, we ca.n write the stress relationship in matrix form as

au au

aU

^{au aw au}

A, - + A , -

ax ax ax ay ax

az

A , ^- + A , - A, - + A , ^-

au

au av au

^{a w}

au

A, - + A , -

ay ax

au

^{aY aY} a t

A , , - + A , - A , - + A , -

au aw

av

^{aw aw aw}

A , - + A , -

az

ax

az

aY

az az

A , - + A , ^- A, - + A , ^-

.

(3.53)

Now the stress itself can vary throughout the fluid and can similarly be expanded in a series, so that the change in rV: across d z is (&,/az) d z , as in Fig. 3.16, for instance. By considering the fluid element of Fig. 3.17, the

a _f

X Y

Fig. 3.17 Deformation of elemental fluid volume with time under shear stresses, shown schematically. For incompressible flow, both the mass and the volume of the element remain constant during the deformation.

Hydrodynamic Equations of the Sea 97

net viscous force across all six faces in the x direction is seen to be given by the sum

and the force per unit mass, d 3 F x / p d x dy d z , is

= - _P 1

(ax

^{a 7 x x}

+ + ”> _az ^.

^(3.55)

Note that this summation proceeds down columns in Eq. 3.53. Repeating the sums for t h e y and

z

components, we obtain the vector form for the vis- cous body force:

1 P

f . V I S = - V ‘ T . (3.56) Here the divergence of the stress tensor has the matrix representation as:

= ( V * A - V ) u

+

( A * V ) V * u

+ [ ( a x + ay + a,)

T r ( A ) ] v ( u

+

^u

+

^{W )}

.

^(3.57)

The meaning of the various differential operators in Eq. 3.57 may be de- rived by taking the gradient of the matrix elements of Eq. 3.53, and then grouping the results into three expressions, as shown in Eq. 3.57. The first of these arises from the left-hand term in the matrix elements given by Eq.

3.52, and the remaining two come for the right-hand term. Discussing them in that order, they represent (1) the effects of eddy viscosity interacting with fluid shear, ( V . A - V)U; (2) the effects of compressibility, as represented by ( A . V ) V . u (which we will discuss in more detail ahead); and (3) the effects of spatial variations in the eddy viscosity as manifested by the expression

[(a, + a, +

d,)Tr(A)]v(u

+

^u

+

^{w), where Tr (A)} is the trace, or sum of diagonal elements of A. The first of these is by far the most important in geophysical fluids, and in fact, for an incompressible, uniform fluid, the last two are zero. For the remainder of this book (except for the chapter on acoustics), we shall assume that the fluid is incompressible, except for the static compression resulting from the weight of the overlying water. We shall also assume that the eddy viscosity does not vary in space. Then the sole remaining contribution to the viscous body force is

1 P

f,,, = - V . A * V U

[:I

1 P

= - [d,A,d,

+

ayAYdy

+

^{d,A,d,] (3.58)}

With these assumptions, the complex sequence of differentiations of a ten- sor has contracted to a comparatively simple scalar operator working on each component of velocity, the expression for which can be written in the suc- cinct form of Eq. 3.58. It is clear that the viscous forces involve second deriva- tives of velocity; to the extent that the coefficients Ai do not vary, those forces are proportional to the curvature of the velocity field and to the mag- nitude of the coefficients of viscosity. Thus viscosities are most important where the velocity field varies most rapidly in its second derivatives, which generally occurs in boundary layers.

In the ocean, the eddy coefficients for horizontal motions are much great- er than for vertical movement. In the horizontal they are often but not al- ways isotropic, although in general they are dependent on horizontal position.

If we call the horizontal and vertical eddy coefficients A , and A , , respec- tively, then the total viscosities may be written as

A, = A , = A ,

+

^{p (3.59a)}

and

A , = A , + p . (3.59b)

Hydrodynamic Equations of the Sea

Thus the tensor form of the eddy and molecular viscosities becomes

99

0

A =

2 B]+

^{p i , (3.60)}

where I is the unit 3 x 3 diagonal matrix. If, in addition, the eddy coeffi- cients are independent of position, the viscous operator simplifies to

where the horizontal Laplacian is

a2 a 2

ax2 ay2

v 2 = - + - , h

and the quantity

v,n = LL - P

(3.61)

(3.62)

(3.63) is termed the kinematic viscosity, which may be regarded as the molecular momentum diffusivity of the fluid. Its value in seawater at 20°C and 35 psu is 1.049 X 10

-'

^{m2 s - I .} The new quantities Kh and K , are analogous to the molecular kinematic viscosity and are termed the eddy dij'fusivities for momentum. Thus

and

K , --= A , / p

.

(3.64b)

Some representative values for A h / p and A , / p are listed in Table 3.2; the lower values go with smaller-scale motions, and conversely.

The dimensions of the diffusion coefficients (square meters per second) reveal something of the physics they describe, i.e., the mean-square step size of random diffusion and mixing processes per unit of time. It will become clear later that the horizontal dimensions of most orderly large-scale mo- tions of the ocean (10 to 1000 km) are much greater than the vertical mo- tions (0.01 to 100 m), and that the random motions must follow suit. The

reasons for choosing anisotropic eddy diffusion coefficients follow from two of the properties of the sea discussed earlier: (1) its essential shallowness when compared with the horizontal scales of many of its motions, and (2) the strong vertical stability that the ocean generally possesses. Both of these charac- teristics make mixing and diffusion in the vertical much smaller than in the horizontal.

TABLE 3.2 Typical Values of Eddy Diffusion Coefficients

K , = A , / p = 3 x lo-’ to 2 x m’ s - ’

Kh = A,/p = 10’ to lo5 m’ s - ’

In summary, the internal body forces in the ocean are modeled by pres- sure and eddy viscosity, viz:

(3.65) where the second term is given by Eq. 3.61.

3.10 Conservation Equations for Mass and Salinity