Environmental Fluid Mechanics - Chapter 13 pot

Topics in Stratified Flow Nearly all natural surface water bodies are stratified at least part of the time.This means there are density variations, usually in the vertical direction.Horizo

Topics in Stratified Flow

Nearly all natural surface water bodies are stratified at least part of the time.This means there are density variations, usually in the vertical direction.Horizontal variations also may exist, but not in steady state unless there areother forces such as Coriolis effects present to balance the resulting pressuredifferences (see Chap 9) Density variations exist most commonly because

of temperature and/or salinity gradients Salinity is the main contributor todensity in the oceans and in some inland lakes such as the Great Salt Lake

in Utah or the Dead Sea in Israel, though these water bodies also may havetemperature gradients In freshwater lakes, temperature stratification is mostimportant In fact, from a water quality point of view, there is usually greatinterest in modeling the temperature structure of water bodies Most biologicaland chemical reactions depend on temperature, and fish choose habitats basedpartly on this parameter As discussed in Chap 12, gas transfer across theair/water interface also depends on temperature

A closely related parameter to density is buoyancy, defined as

b D g
0

0 D g0

13.1.1

where
is the density of a fluid parcel and
0 is the reference density

Buoy-ancy is also known as reduced gravity, g0, and is defined so that a fluid parceltends to rise when its buoyancy is positive (i.e., its density is less than thereference value) and a particle with higher buoyancy has a greater tendency torise Buoyancy may be treated like other state variables, such as temperature

or concentration, and a conservation equation can be defined,

where kb is diffusivity for buoyancy and the source/sink terms are defineddepending on which component is contributing to the buoyancy For example,

a solar heating term might generate buoyancy in a temperature-stratified system(Sect 12.2), while evaporation might cause (negative) buoyancy at the surface,due either to cooling (heat loss due to evaporation) or to increased salinity in

the case of saline water Usually, an equation of state linking density or

buoy-ancy to these other properties is needed in addition to the general equations

of motion in order to define these systems

13.1.1 Equation of State

Density is related to temperature and salinity and possibly other properties of

a particular system through an equation of state A well-known example of

such an equation is the perfect gas law, which relates density, pressure, and

temperature through the gas constant Similarly, a general equation of statemay be formulated for water systems as

where T D temperature, C D concentration of dissolved species, and p Dpressure Since the main stratifying agent for most natural systems, in terms

of dissolved species, is salinity S, this will be used in place of C Also, except

in the deep oceans or the atmosphere, the incompressible assumption impliesthat density should not be a function of pressure Thus

A number of expressions have been proposed to define this relationship,usually based on high-order polynomial fits to tabulated values of density as afunction of T and S Relations have been proposed for simple sodium chloridesolutions and also to actual seawater solutions In general the dependence ontemperature has been found to be approximately parabolic, with a maximum at

4°C (Fig 13.1) However, the temperature of the density maximum changeswith increasing salinity, decreasing to about 0°C for highly saline systems

To a first-order approximation, density is linearly dependent on salinity overmuch of the normal range of interest

The dependence of density on T and S is expressed through values ofthe thermal and saline expansion coefficients, ˛ and ˇ, respectively, where

Figure 13.1 Variation of freshwater density with temperature.

The negative sign in the definition for ˛ is meant so that, at least for T > 4°C,

the coefficient has a positive value Note that ˛ changes sign for temperaturesless than the temperature of the density maximum In general, both ˛ and ˇare functions of T and S, but approximate values are ˛ ¾D 2 ð 10
4 °C
1 and

ˇ ¾D 7.5 ð 10
3S
1, where S is in weight percent It is clear from these valuesthat salinity has a much greater effect on density than does temperature.The equation of state is written to express density in terms of ˛ and ˇ,and deviations of T and S from standard or reference values as

where T D T
T0, S D S
S0,
D
0 when T D T0 and S D S0, and

T0 and S0 are the reference values for temperature and salinity, respectively.Normally, T0D 4°C and S0 D 0 (%) Note that S is usually expressed inunits of weight percent, parts per thousand, or mg/L Typical seawater has

S ¾D 3.5%, 35 ppt (o/oo), or 35,000 ppm When ˛ and ˇ are taken as constants,

the resulting equation is called a linear equation of state A constant value for

ˇis usually a good approximation, but in order to account for the parabolicnature of the temperature dependence, ˛ should be a function of T A simpleexpression that gives reasonable results over much of the range of normally

occurring values for T and S (except near freezing) is

where T is in°C and S is in weight percent For freshwater bodies, with S D 0,this equation is reasonable even near freezing When S is high, however,greater than about 5%, higher order equations should be used to estimate

13.1.2 Gravitational Stability

When density differences exist in a fluid system, an important tion is that of stability In Chap 10the concept of convective transport wasintroduced, where it was noted that convection is the result of a gravitation-ally unstable condition This is simply saying that lighter, more buoyant fluidshould tend to rise and that the system would be stable only when heavier fluidunderlies lighter fluid Buoyancy instabilities give rise to convective motions,which tend to mix the fluid system vertically This is demonstrated mathemat-

considera-ically by applying a perturbation analysis to a system in which there is no

motion initially (u DvD w D 0), with a density stratification
z, as trated inFig 13.2.A fluid particle is considered with initial position at z D 0,which is an arbitrary point within the fluid

illus-Figure 13.2 Fluid particle at z D 0, in fluid with ambient stratification given by

z D
r(reference distribution).

The governing equation for this problem is



0 D
1

This is simply the momentum equation for the vertical direction, which reduces

to the hydrostatic result for the case of zero flow The reference density,
r,must satisfy this equation An equation of state also should be considered

to relate density to other properties of the system, as noted above The fluid

is assumed to be incompressible (note that an incompressible fluid does nothave to have the same density everywhere) If the fluid particle ofFig 13.2isdisplaced vertically from its equilibrium location by an amount z, there will

be a buoyancy force acting on the particle due to the difference in densitybetween the particle and its new surroundings The resulting force acting onthe particle, per unit volume, is [
g
p

r], where
p is the fluid particledensity Applying Newton’s law,

Taylor series expansions are now defined for
r and
p, in terms of areference value
0:

no heat transfer, the relationship between pressure and density is given bythe adiabatic relationship, dp D c2d
, where c D sonic velocity Makingthese substitutions, along with the hydrostatic pressure result (13.1.9), into

The Brunt–Vaisala or buoyancy frequency, N, is defined in terms of the

square root of the negative of the term in parentheses in this last expression.Note that the density gradient is assumed to be negative, as it must be for

a gravitationally stable water column; N is undefined for an unstable densitydistribution However, the sonic velocity c is normally large, and the first term

in parentheses on the right-hand side of Eq (13.1.4) can be neglected undermost conditions This is equivalent to neglecting the small compressibility ofthe fluid parcel (resulting in a small change in density) due to the change inambient pressure at the perturbed location In general, N is defined for mostapplications by

where it has been assumed that
p¾D

0 The final differential equation is

which is an equation of simple harmonic motion

Equation (13.1.17) has solutions of the form

on the fluid particle when it is displaced from its original equilibrium position,and particle position remains constant For the stable situation, substituting theEuler formula,

it is seen that oscillatory motions are expected, with amplitude depending

on the magnitude of the original displacement of the fluid particle Theseoscillations decay over time due to viscous effects, which have not beenconsidered here but could be included in the equation of motion for the fluid

particle, Eq (13.1.17), if desired For the unstable case, the particle positiongrows exponentially with time Thus any perturbation of particle position in

an unstable environment will lead to large-scale convective motions

We now consider wave motions that are possible in a stratified fluid Internalwaves can propagate along the interface between fluid layers of different densi-ties (note that surface waves, as discussed in Chap 8, propagate along theair/water interface, which is an extreme example of fluids of two differentdensities) or, more generally, at an angle to the horizontal through a densitystratified fluid, with N2 >0

Consider a stratified fluid with density and pressure fields given by

of
0, which in turn is much greater than the magnitude of
00 (one or twoorders of magnitude difference between each component) Since we are dealingprimarily with water, incompressibility dictates that the density following afluid particle is constant, or

For the analysis of stratified fluids it is convenient to consider a reference state

of zero motion The reference density is
rD 
0C
0z(seeFig 13.2).Thepressure is given similarly as prD p0C p0z Substituting this definition for

r into Eq (13.2.3) results in

Then, using the definitions for pr and
r, along with the approximation that

1/
 ¾D 1/
0(this is the Boussinesq approximation discussed in Sect 2–7),

where uh denotes a horizontal velocity component, i.e., h takes values of 1

or 2, for the x or y direction, respectively The fact that p006D fz has alsobeen taken into account in writing Eq (13.2.11) Taking the derivative of

Eq (13.2.10) with respect to z and subtracting the derivative of Eq (13.2.11)with respect to xh gives

This last result is then differentiated with respect to t and the two componentequations (for h D 1 or 2) are then

The right-hand side is generally negligible compared with the other terms

in the equation, since it involves all nonlinear terms A possible exception tothis is when there is strong mean shear, in which case the velocity gradientsmay be important The simplified final equation is

∂2

which is a wave equation similar to Eq (13.1.17)

It is beyond the scope of the present text to provide a full discussion

of the solutions to this equation (or the more general Eq 13.2.19) and theresulting behavior of the fluid motions that it describes There are many booksthat cover this material in depth, and the present analysis is restricted to adiscussion of some properties of linear internal waves and their relationship tosurface waves First, however, we consider the lowest mode solutions, whichcorrespond to horizontal propagation

13.2.1 Lowest Mode Solutions

By assuming wavelike disturbances,

at the surface and at the bottom

The free surface is defined by x, y, t, as sketched inFig 13.3.At thesurface, z D , the dynamic boundary condition is

and the kinematic boundary condition is

D

However, application of boundary conditions is problematic at z D , since

 itself is an unknown, obtained as part of the solution For a first-ordersolution, it is more convenient to write the surface boundary conditions at

z D  D0 and to account for variations between this level and the actual

Figure 13.3 Definition sketch for surface level.

surface through linear Taylor series expansions of Eqs (13.2.23) and (13.2.24),assuming approximately hydrostatic pressure variations and small  Then, for

z D0, the dynamic condition is

Substituting this last result into Eq (13.2.28), along with the boundary tion (13.2.27), gives

From examination of the governing Eq (13.2.22), if N2 2, then W

is monotonic in zW / expjjz, where 2D N2 2 2 <0 — recall theearlier discussion of gravitational stability in Sec 13.1) In this case, themagnitude of the motions decays exponentially with depth, as z becomesmore negative A limiting case is when N D 0 (no stratification), where onlysurface waves can exist In fact, surface waves are thus seen as a special

max, where Nmax is themaximum value of the buoyancy frequency in the water column, then there

is a range of z over which W and (d2W/dz2) must have opposite signs, inorder that Eq (13.2.22) can be satisfied This is a characteristic of oscillatingfunctions, and it may be concluded that W is oscillating in this region (i.e.,

max, thereare many possible values for kh The lowest value (lowest mode) correspondswith surface waves, and higher modes correspond with internal waves For thelowest internal wave mode, the entire thermocline (or pycnocline) moves upand down in unison As the mode increases, corresponding to larger kh, there

is a shortening of the vertical scale of motions, with more zero-crossings Thevertical scale of these motions is estimated by

and the horizontal scale is simply the inverse of the wave number, LH³ k
1h ,

so, using Eq (13.2.22),

D N, Lz× LH, and variations in the vertical flow field are small; this isthe case for the lowest mode of internal wave.

For this lowest internal mode, consider the situation where the densitystratification approaches a step function, as in the two-layer stratificationillustrated in Fig 13.4 Here, N D 0 everywhere except at the densityinterface at z D
h, where it is large The interface has thickness υ, which

is small compared with h In the uniform regions, the governing waveequation (13.2.22) reduces to

Figure 13.4 Distribution of density and buoyancy frequency in a two-layer stratified system.

The full equation (13.2.22), integrated over the interfacial thickness υ, givesdW

The solution of (13.2.34), with the stated boundary conditions, isWz D

2¾ gυ



0

13.2.40For the other extreme, khυ −1, which is more commonly the case, and

Surface Manifestation of Lowest Internal Wave Mode

One of the most important effects of the lowest internal wave mode is itsrelationship to surface waves This results from the pressure field created bythe internal wave Consider a wave function describing the motions at theinterface, given by

where A is the amplitude of the motions Using a similar approach as indescribing the boundary condition at z D  (see text following Eq 13.2.24),the vertical velocity at the interface is

At the surface, the linearized horizontal momentum equation is (from

Eq 13.2.9, without the viscous term)

Upon comparing this result with Eq (13.2.43), the magnitude of  relative to

suggests that the surface “signature” of the internal wave is relatively small,compared with the amplitude of the internal wave itself

The phase speed of the waves is defined as the ratio of wavelength to

13.2.2 Small-Scale Internal Waves

The preceding discussion focused on the lowest internal wave mode and itsrelationship with surface waves The analysis is now generalized to considerhigher mode internal waves, but with the assumption that the vertical scale

of motion is small relative to the scale over which N varies, so that at leastlocally, it may be assumed that N is approximately constant The equationdescribing vertical velocity is Eq (13.2.20), with solutions assumed to be of aform similar to Eq (13.2.21) but generalized to include a vertical component

of the wave number vector,

where is the angle between the horizontal and the total wave number vector,

as shown inFig 13.5,and cos D kh/kk2D k2

1 C k2

2C k2

3 This is the angle

of wave propagation, relative to horizontal

maxj D N, which corresponds with horizontal wave propagation (khD k)

hD 0 It also is of interest tonote that, in general, the velocity field can be represented by

Figure 13.5 Definition sketch for wave propagation angle

(refer to Eq 13.2.55) For incompressible fluid, the divergence of the velocity

is 0, so

which implies that the wave number vector and the velocity vector areperpendicular to each other This is in contrast to surface waves, where thevelocity and wave number vectors are in the same direction A further result

of Eq (13.2.58) is

which shows that the linearized solution is an exact solution to the governingequations, since the neglected nonlinear terms in the linearized solution are infact zero

13.3.1 Linear Stability Theory

One of the consequences of internal wave propagation is the transport ofenergy, and under certain circumstances waves may break and release energythat can be used for mixing, which is a topic of considerable interest instratified flow modeling One of the more well-known modes of this type

of mixing is a process known as Kelvin–Helmholtz instability, which refers

to instability and breaking of an interfacial wave First, however, considerthe general stability problem for a system with a steady, two-dimensional,inviscid, mean shear flow Uz Hydrostatic pressure is assumed, and the meanvelocity field,u D[Uz, 0, 0] satisfies the exact (nonlinear) two-dimensional

governing equations Perturbations to this mean velocity field are assumed as

u D U C u0, w0 p D p0C p0

D
0C
0

13.3.1where primes indicate perturbation quantities and p0 and
0 are the pressureand density, respectively, of the unperturbed system; these are both functions

∂u0

∂x C∂w0

This is similar to the behavior of the turbulent velocity fluctuations described

inChap 5.The momentum equations are (neglecting viscous effects)

Subtracting the derivative of Eq (13.3.5) with respect to z from the derivative

of Eq (13.3.6) with respect to x eliminates the pressure terms, and

A solution is found by assuming wavelike expressions,

and for the case N D 0, this is known as the Rayleigh equation This last

result forms an eigenvalue problem for c If the imaginary part of c is greaterthan 0, the perturbations will grow exponentially and the flow is dynamicallyunstable

To develop the result for Kelvin–Helmholtz instability, consider a

two-layer system, with z D 0 defined at the interface between the two two-layersand with densities and velocities
1 and U1, and
2 and U2, in the upperand lower layers, respectively (Fig 13.6) Following the preceding analysis,vertical velocity w0 is assumed to be described by an equation of the form

of Eq (13.3.8) Within each layer, N D 0 and U is constant, so Eq (13.3.12)reduces to

d2Ow

Figure 13.6 Definition sketch for stability analysis of two-layer system.

(note the similarity of this result to Eq 13.2.34) The solution is an tial, and in order to keep the velocities bounded as z ! š1, we have

Then, combining Eqs (13.3.14) and (13.3.15) gives

Ow2jzD0
D ikU2
cO D C2 13.3.17bSince these must be equal, a relationship between C1 and C2 is obtained,

1ikU1
cu1 0C
2ikU2
cu2 0D 
1

2gik O 13.3.23Using Eq (13.3.10), along with Eq (13.3.14) to evaluate dw0/dzat the inter-face (z D 0), we find

u1 0D ik

into Eq (13.3.23), along with

Eq (13.3.17), assuming C16D 0 and rearranging, we obtain


1C
2c2
2
1U1C
2U2c C 
1U21C
2U22

2

1g

k D 0

13.3.25

This last result is a simple quadratic equation for c, which when solvedprovides a dispersion relationship,

c D š

gk



2

13.3.28where it has been assumed that 
D
2

1 is small relative to either
1

or
2

The stability condition is usually written in terms of an interfacial

Richardson number,

RiiD gk
2



so that when Rii<0.5, the system is unstable In other words, for a given

U, there exists a k large enough that instability occurs, as an tially growing wave If 
D 0, any perturbation is always unstable, since

exponen-Eq (13.3.28) will be satisfied for any U

A generalized stability condition was derived by Rayleigh, based on avariation of the Taylor–Goldstein equation (13.3.11),

ddz



0

dUdz



0

dUdz

u1 0D ik

into Eq (13. 3.23),... constant, so Eq (13. 3.12)reduces to

d2Ow

Figure 13. 6... we have

Then, combining Eqs (13. 3.14) and (13. 3.15) gives

Ow2jzD0