4.9. Special Forms of the Equations

The general equations of motion for a fluid may be put into a variety of special forms when certain symmetries or approximations are valid. Several special forms are presented in this section. The first applies to the integral form of the momentum equation and corresponds to the classical mechanics principle of conservation of angular momentum. The second through fifth special forms arise from manipulations of the differential equations to generate Bernoulli equations. The sixth special form applies when the flow has constant density and the gravitational body force and hydrostatic pressure cancel. The final special form for the equations of motion presented here, known as the Boussinesq approximation, is for low-speed incompressible flows with constant transport coefficients and small changes in density.

Angular Momentum Principle for a Stationary Control Volume

In the mechanics of solid bodies it is shown that

dH/dt=M,

(4.64)

where M is the torque of all external forces on the body about any chosen axis, and dH/dt is the rate of change of angular momentum of the body about the same axis. For the fluid in a material control volume, the angular momentum is

H=∫V(t)(r×ρu)dV,

where r is the position vector from the chosen axis (Figure 4.11). Inserting this in (4.64) produces:

ddt∫V(t)(r×ρu)dV=∫V(t)(r×ρg)dV+∫A(t)(r×f)dA,

where the two terms on the right are the torque produced by body forces and surface stresses, respectively. As before, the left-hand term can be expanded via Reynolds transport theorem to find:

Figure 4.11 Definition sketch for the angular momentum theorem where dm = ρdV. Here the chosen axis points out of the page, and elemental contributions the angular momentum about this axis are r×ρudV.

ddt∫V(t)(r×ρu)dV=∫V(t)∂∂t(r×ρu)dV+∫A(t)(r×ρu)(u·n)dA=∫Vo∂∂t(r×ρu)dV+∫Ao(r×ρu)(u·n)dA=ddt∫Vo(r×ρu)dV+∫Ao(r×ρu)(u·n)dA,

where V_o and A_o are the volume and surface of a stationary control volume that is instantaneously coincident with the material volume, and the final equality holds because V_o does not vary with time. Thus, the stationary volume angular momentum principle is:

ddt∫Vo(r×ρu)dV+∫Ao(r×ρu)(u·n)dA=∫Vo(r×ρg)dV+∫Ao(r×f)dA.

(4.65)

The angular momentum principle (4.65) is analogous to the linear momentum principle (4.17) when b = 0, and is very useful in investigating rotating fluid systems such as turbomachines, fluid couplings, dishwashing-machine spray rotors, and even lawn sprinklers.

Example 4.12

Consider a lawn sprinkler as shown in Figure 4.12. The area of each nozzle exit is A, and the jet velocity is U. Find the torque required to hold the rotor stationary.

Figure 4.12 Lawn sprinkler.

Solution

Select a stationary volume V_o with area A_o as shown by the dashed lines. Pressure everywhere on the control surface is atmospheric, and there is no net moment due to the pressure forces. The control surface cuts through the vertical support and the torque M exerted by the support on the sprinkler arm is the only torque acting on V_o. Apply the angular momentum balance

∫Ao(r×ρu)(u·n)dA=∫Ao(r×f)dA=M,

 

where the time derivative term must be zero for a stationary rotor. Evaluating the surface flux terms produces:

∫Ao(r×ρu)(u·n)dA=(aρUcosα)UA+(aρUcosα)UA=2aρAU2cosα.

 

Therefore, the torque required to hold the rotor stationary is M=2aρAU2cosα. When the sprinkler is rotating at a steady rate, this torque is balanced by air resistance and mechanical friction.

Bernoulli Equations

Various conservation laws for mass, momentum, energy, and entropy were presented in the preceding sections. Bernoulli equations are not separate laws, but are instead derived from the Navier-Stokes momentum equation (4.38) or the energy equation (4.60) under various sets of conditions.

First consider inviscid flow (μ = μ_υ = 0) where gravity is the only body force so that (4.38) reduces to the Euler equation (4.41):

∂uj∂t+ui∂uj∂xi=−1ρ∂p∂xj−∂∂xjΦ,

(4.66)

where Φ = gz is the body force potential, g is the acceleration of gravity, and the z-axis is vertical. If the flow is also barotropic, then ρ = ρ(p), and the pressure gradient term can be rewritten in terms of an integral:

∂∂xj∫popdp′ρ(p′)=∂∂p(∫popdp′ρ(p′))∂p∂xj=1ρ∂p∂xj,

(4.67)

where dp/ρ is a perfect differential, p_o is a reference pressure, p′ is the integration variable, the middle expression follows from the chain-rule for partial differentiation, and the final one follows from the rules for differentiating an integral with respect to its upper limit. When ρ = ρ(p), the integral in (4.67) depends only on its endpoints, and not on the path of integration. Constant density, isothermal, and isentropic flows are barotropic. In addition, the advective acceleration in (4.66) may be rewritten in terms of the velocity-vorticity cross product, and the gradient of the kinetic energy per unit mass:

ui∂uj∂xi=−εikjuiωk+∂∂xj(12ui2),or(u·∇)u=−u×ω+∇(|u|2/2)

(4.68)

(see Exercise 4.58). Substituting (4.67) and (4.68) into (4.66) produces:

∂u∂t+∇[12|u|2+∫popdp′ρ(p′)+gz]=u×ω,

(4.69)

where all the gradient terms have been collected together to form the Bernoulli function B = the contents of the [,]-brackets.

Equation (4.69) can be used to deduce the evolution of the Bernoulli function in inviscid barotropic flow. First consider steady flow (∂u/∂t=0) so that (4.69) reduces to

∇B=u×ω.

(4.70)

The left-hand side is a vector normal to the surface B = constant whereas the right-hand side is a vector perpendicular to both u and ω (Figure 4.13). It follows that surfaces of constant B must contain the streamlines and vortex lines. Thus, an inviscid, steady, barotropic flow satisfies:

12|u|2+∫popdp′ρ(p′)+gz=constantalongstreamlinesandvortexlines.

(4.71)

This is the first of several possible Bernoulli equations. If, in addition, the flow is irrotational (ω = 0), then (4.70) implies that

12|u|2+∫popdp′ρ(p′)+gz=constanteverywhere.

(4.72)

It may be shown that a sufficient condition for the existence of the surfaces containing streamlines and vortex lines is that the flow be barotropic. Incidentally, these are called Lamb surfaces in honor of the distinguished English applied mathematician and hydrodynamicist, Horace Lamb. In a general nonbarotropic flow, a path composed of streamline and vortex line segments can be drawn between any two points in a flow field. Then (4.71) is valid with the proviso that the integral be evaluated on the specific path chosen. As written, (4.71) requires that the flow be steady, inviscid, and have only gravity (or other conservative) body forces acting upon it. Irrotational flow is presented in Chapter 7. We shall note only the important point here that, in a nonrotating frame of reference, barotropic irrotational flows remain irrotational if viscous effects are negligible. Consider the flow around a solid object, say an airfoil (Figure 4.14). The flow is irrotational at all points outside the thin viscous layer close to the surface of the object. This is because a particle P on a streamline outside the viscous layer started from some point S, where the flow is uniform and consequently irrotational. The Bernoulli equation (4.72) is therefore satisfied everywhere outside the viscous layer in this example.

Figure 4.13 A surface defined by streamlines and vortex lines. Within this surface the Bernoulli function defined as the contents of the [,]-brackets in (4.69) is constant in steady flow. Note that the streamlines and vortex lines can be at an arbitrary angle.

Figure 4.14 Flow over a solid object. Viscous shear stresses are usually confined to a thin layer near the body called a boundary layer. Flow outside the boundary layer is irrotational, so if a fluid particle at S is initially irrotational it will remain irrotational at P because the streamline it travels on does not enter the boundary layer.

An unsteady form of Bernoulli’s equation can be derived only if the flow is irrotational. In this case, the velocity vector can be written as the gradient of a scalar potential ϕ (called the velocity potential):

u≡∇ϕ.

(4.73)

Putting (4.73) into (4.69) with ω = 0 produces:

∇[∂ϕ∂t+12|∇ϕ|2+∫popdp′ρ(p′)+gz]=0,or∂ϕ∂t+12|∇ϕ|2+∫popdp′ρ(p′)+gz=B(t),

(4.74)

where the integration function B(t) is independent of location. Here ϕ can be redefined to include B,

ϕ→ϕ+∫totB(t′)dt′,

without changing its use in (4.73). Then, the second part of (4.74) provides a second Bernoulli equation for unsteady, inviscid, irrotational, barotropic flow:

∂ϕ∂t+12|∇ϕ|2+∫popdp′ρ(p′)+gz=constant.

(4.75)

This form of the Bernoulli equation will be used in studying irrotational wave motions in Chapter 8.

A third Bernoulli equation can be obtained for steady flow (∂/∂t = 0) from the energy equation (4.55) in the absence of viscous stresses and heat transfer (τ_ij = q_i = 0):

ρui∂∂xi(e+12uj2)=ρuigi−∂∂xj(ρujp/ρ).

(4.76)

When the body force is conservative with potential gz, and the steady continuity equation, ∂(ρu_i)/∂x_i = 0, is used to simplify (4.76), it becomes:

ρui∂∂xi(e+pρ+12uj2+gz)=0.

(4.77)

From (1.19) h = e + p/ρ, so (4.77) states that gradients of the sum h + |u|²/2 + gz must be normal to the local streamline direction u_i. Therefore, a third Bernoulli equation is:

h+12|u|2+gz=constantonstreamlines.

(4.78)

This result is consistent with the momentum Bernoulli equations (4.71), (4.72), and (4.75). Equation (4.63) requires that inviscid, nonheat-conducting flows are isentropic (s does not change along particle paths), and (1.24) implies dp/ρ = dh when s = constant. Thus the path integral ∫dp/ρ in (4.71), (4.72), and (4.75) becomes a function h of the endpoints only if both heat conduction and viscous stresses may be neglected. Equation (4.78) is very useful for high-speed gas flows where there is significant interplay between kinetic and thermal energies along a streamline. It is nearly the same as (4.71), but does not include the other barotropic and vortex-line-evaluation possibilities allowed by (4.71).

Interestingly, there is also a Bernoulli equation for constant-viscosity constant-density irrotational flow. It can be obtained by starting from (4.39), using (4.68) for the advective acceleration, and noting from (4.40) that ∇2u=−∇×ω in incompressible flow:

ρDuDt=ρ∂u∂t+ρ∇(12|u|2)−ρu×ω=−∇p+ρg+μ∇2u=−∇p+ρg−μ∇×ω.

(4.79)

When ρ = constant, g=−∇(gz), and ω = 0, the second and final parts of this extended equality require:

ρ∂u∂t+∇(12ρ|u|2+ρgz+p)=0.

(4.80)

Now, form the dot product of this equation with the arc-length element e_uds = ds directed along a chosen streamline, integrate from location 1 to location 2 along this streamline, and recognize that eu·∇=∂/∂s to find:

ρ∫12∂u∂t·ds+∫12eu·∇(12ρ|u|2+ρgz+p)ds=ρ∫12∂u∂t·ds+∫12∂∂s(12ρ|u|2+ρgz+p)ds=0.

(4.81)

The integration in the second term is elementary, so a fourth Bernoulli equation for constant-viscosity constant-density irrotational flow is:

∫12∂u∂t·ds+(12|u|2+gz+pρ)2=(12|u|2+gz+pρ)1,

(4.82)

where 1 and 2 denote upstream and downstream locations on the same streamline at a single instant in time. Alternatively, (4.80) can be written using (4.73) as:

∂ϕ∂t+12|∇ϕ|2+gz+pρ=constant.

(4.83)

To summarize, there are (at least) four Bernoulli equations: (4.71) is for inviscid, steady, barotropic flow; (4.75) is for inviscid, irrotational unsteady, barotropic flow, (4.78) is for inviscid, isentropic, steady flow, and (4.82) or (4.83) are for constant-viscosity, irrotational, unsteady, constant density flow. Perhaps the simplest form of these is (4.19).

There are many useful and important applications of Bernoulli equations. A few of these are described in the following paragraphs.

Consider first a simple device to measure the local velocity in a fluid stream by inserting a narrow bent tube (Figure 4.15), called a pitot tube after the French mathematician Henri Pitot (1695–1771), who used a bent glass tube to measure the velocity of the river Seine. Consider two points (1 and 2) at the same level, point 1 being away from the tube and point 2 being immediately in front of the open end where the fluid velocity u₂ is zero. If the flow is steady and irrotational with constant density along the streamline that connects 1 and 2, then (4.19) gives:

Figure 4.15 Pitot tube for measuring velocity in a duct. The first port measures the static pressure while the second port measures the static and dynamic pressure. Using the steady Bernoulli equation for incompressible flow, the height difference h₂ – h₁ can be related to the flow speed.

p1ρ+12|u1|2=p2ρ+12|u2|2=p2ρ

from which the magnitude of u1 is found to be:

|u1|=2(p2−p1)/ρ.

Pressures at the two points are found from the hydrostatic balance:

p1=ρgh1andp2=ρgh2,

so that the magnitude of u₁ can be found from:

|u1|=2g(h2−h1).

Because it is assumed that the fluid density is very much greater than that of the atmosphere to which the tubes are exposed, the pressures at the tops of the two fluid columns are assumed to be the same. They will actually differ by ρ_atmg(h₂ − h₁). Use of the hydrostatic approximation above station 1 is valid when the streamlines are straight and parallel between station 1 and the upper wall.

The pressure p₂ measured by a pitot tube is called stagnation pressure or total pressure, which is larger than the local static pressure. Even when there is no pitot tube to measure the stagnation pressure, it is customary to refer to the local value of the quantity (p + ρ|u|²/2) as the local stagnation pressure, defined as the pressure that would be reached if the local flow is imagined to slow down to zero velocity frictionlessly. The quantity ρu²/2 is sometimes called the dynamic pressure; stagnation pressure is the sum of static and dynamic pressures.

As another application of Bernoulli’s equation, consider the flow through an orifice or opening in a tank (Figure 4.16). The flow is slightly unsteady due to lowering of the water level in the tank, but this effect is small if the tank area is large compared to the orifice area. Viscous effects are negligible everywhere away from the walls of the tank. All streamlines can be traced back to the free surface in the tank, where they have the same value of the Bernoulli constant B = |u|²/2 + p/ρ + gz. It follows that the flow is irrotational, and B is constant throughout the flow.

Figure 4.16 Flow through a sharp-edged orifice. The pressure is atmospheric everywhere across section CC; its distribution across orifice AA is indicated. The basic finding here is that the hole from which the jet emerges is larger than the width of the jet that crosses CC.

Application of the Bernoulli equation (4.19) for steady constant-density flow between a point on the free surface in the tank and a point in the jet downstream of CC gives:

patmρ+gh=patmρ+u22,

from which the average jet velocity magnitude u is found as:

u=2gh,

which simply states that the loss of potential energy equals the gain of kinetic energy.

To recover the mass flow, the jet's cross sectional area is needed. However, the conditions right at the opening (section AA in Figure 4.16) are not simple because the pressure is not uniform across the jet outlet and streamlines are curved. Although pressure has the atmospheric value everywhere on the free surface of the jet (neglecting small surface tension effects), it is not equal to the atmospheric pressure inside the jet at the section AA. The curved streamlines at the orifice indicate that pressure must increase toward the centerline of the jet to balance the centrifugal force. A sketch of the pressure distribution across the orifice (section AA) is shown in Figure 4.16. However, the streamlines in the jet become parallel a short distance away from the orifice (section CC in Figure 4.16), where the jet area is smaller than the orifice area. The pressure across section CC is uniform and equal to the atmospheric value (p_atm). Thus, the mass flow rate in the jet is approximately: m˙=ρAcu=ρAc2gh, where A_c is the area of the jet at CC. For orifices having a sharp edge, A_c has been found to be ≈62% of the orifice area because the jet contracts downstream of the orifice opening.

If the orifice has a well-rounded opening (Figure 4.17), then the jet does not contract, the streamlines right at the exit are parallel, and the pressure at the exit is uniform and equal to the atmospheric pressure. Consequently the mass flow rate is simply ρA2gh, where A is the orifice area. Thus, a simple way to increase the flow rate from such an orifice is to provide a well-rounded opening.

Figure 4.17 Flow through a rounded orifice. Here the pressure and velocity can achieve parallel outflow inside the tank, so the width of the jet does not change outside the tank.

Example 4.13

A perfect gas with specific heat ratio γ at temperature T₁ escapes horizontally at the speed of sound (u₁ = c) from a small leak in a pressure vessel (Figure 4.18). What is the temperature T₂ of the gas as it leaves the pressure vessel if its enthalpy is proportional to T (i.e., h = c_pT) and the flow is steady and frictionless?

Figure 4.18 Pressure vessel with a small leak.

Solution

Start with the steady compressible-flow Bernoulli equation (4.78) without the gravity term and use (1.33), c² = γRT, for the speed of sound c:

h1+12u12=h2+12c2orcpT1+0=cpT2+12γRT2.

 

Here, u₁ ≈ 0 is the speed of the quiescent gas in the pressure vessel, and the sound speed is evaluated at the gas temperature as it passes through the orifice. The second equation is readily solved for T₂:

T2=cpT1/(cp+12γR)=2T1/(γ+1),

 

where (1.29) and (1.30) have been used to reach the final form. Interestingly, this answer does not depend on the pressure in the vessel, and it leads to a noticeable temperature change: T1−T2=[(γ−1)/(γ+1)]T1, which is nearly 50°C for pressurized air starting at room temperature.

Example 4.14

The density and flow speed in the intake manifold of a reciprocating engine are approximately ρ_o (a constant) and u(t) = U_o(1 + sin(2πft)). If the throttle-plate-to-cylinder-intake-valve runner is a straight horizontal tube of length L, (see Figure 4.19) determine a formula for the pressure difference required between the ends of this tube to sustain this fluid motion, assuming frictionless incompressible flow.

Figure 4.19 Simple intake manifold for a reciprocating internal combustion engine.

Solution

Start with the unsteady Bernoulli equation (4.82) and use the tube-centerline streamline between the throttle plate (1) and the intake valve (2). This streamline is the dashed line in Figure 4.19. Several simplifications to (4.82) can be made immediately: the flow is horizontal so the gravity terms do not enter; the fluid velocity is the same at both ends of the tube so the kinetic energy terms cancel; and the dot-product in the acceleration term is readily evaluated (∂u/∂t)·ds=(∂u/∂t)dx. Thus, (4.82) becomes:

∫12∂u∂tdx+p2ρ=p1ρ,orp1−p2=ρ∫12∂u∂tdx=ρ∂u∂t(x2−x1)=2πρfUoLcos(2πft),

(4.82)

 

where x₂ – x₁ = L, and the integral is elementary because ∂u/∂t = 2πfU_ocos(2πft) does not depend on x.

Interestingly, even for a low air density (0.5 kg/m³), a low frequency (50 Hz), a low flow speed (10 m/s), and a short runner length (0.3 m), this estimate produces pressure fluctuations that are enormous from a sound amplitude standpoint: |p₁ – p₂| = 2π(0.5 kg/m³)(50 Hz)(10 m/s)(0.3 m) ≈ 470 Pa. Although this pressure is a tiny fraction of atmospheric pressure, it corresponds to a sound pressure level of more than 140 dB re 20 μPa, a level that quickly causes hearing damage or loss.

Neglect of Gravity in Constant Density Flows

When the flow velocity is zero, the Navier-Stokes momentum equation for incompressible flow (4.39b) reduces to a balance between the hydrostatic pressure p_s, and the steady body force acting on the hydrostatic density ρ_s:

0=−∇ps+ρsg,

which is equivalent to (1.14). When this hydrostatic balance is subtracted from (4.39b), the pressure difference from hydrostatic, p′ = p – p_s, and the density difference from hydrostatic, ρ′ = ρ – ρ_s, appear:

ρDuDt=−∇p′+ρ′g+μ∇2u.

(4.84)

When the fluid density is constant, ρ′ = 0 and the gravitational-body-force term disappears leaving:

ρDuDt=−∇p′+μ∇2u.

(4.85)

Because of this, steady body forces (like gravity) in constant density flow are commonly omitted from the momentum equation, and pressure is measured relative to its local hydrostatic value. Furthermore, the prime on p in (4.85) is typically dropped in this situation. However, when the flow includes a free surface, a fluid-fluid interface across which the density changes, or other variations in density, the gravitational-body-force term should reappear.

The Boussinesq Approximation

For flows satisfying certain conditions, Boussinesq in 1903 suggested that density changes in the fluid can be neglected except where ρ is multiplied by g. This approximation also treats the other properties of the fluid (such as μ, k, c_p) as constants. It is commonly useful for analyzing oceanic and atmospheric flows. Here the basis for the approximation is presented in a somewhat intuitive manner, and the resulting simplifications of the equations of motion are examined. A formal justification, and the conditions under which the Boussinesq approximation holds, is given in Spiegel and Veronis (1960).

The Boussinesq approximation replaces the full continuity equation (4.7) by its incompressible form (4.10), ∇·u=0, to indicate that the relative density changes following a fluid particle, ρ^–1(Dρ/Dt), are small compared to the velocity gradients that compose ∇·u. Thus, the Boussinesq approximation cannot be applied to high-speed gas flows where density variations induced by velocity divergence cannot be neglected (see Section 4.11). Similarly, it cannot be applied when the vertical scale of the flow is so large that hydrostatic pressure variations cause significant changes in density. In a hydrostatic field, the vertical distance over which the density changes become important is of order c²/g ∼ 10 km for air where c is the speed of sound. (This vertical distance estimate is consistent with the scale height of the atmosphere; see Section 1.10.) The Boussinesq approximation therefore requires that the vertical scale of the flow be L ≪ c²/g.

In both cases just mentioned, density variations are caused by pressure variations. Now suppose that such pressure-compressibility effects are small and that density changes are caused by temperature variations alone, as in a thermal convection problem. In this case, the Boussinesq approximation applies when the temperature variations in the flow are small. Assume that ρ changes with T according to δρ/ρ=−αδT, where α = −ρ^–1(∂ρ/∂T)_p is the thermal expansion coefficient (1.26). For a perfect gas at room temperature α = 1/T ∼ 3 × 10^–3 K^–1 but for typical liquids α ∼ 5 × 10^–4 K^–1. Thus, for a temperature difference in the fluid of 10°C, density variations can be at most a few percent, and it turns out that ρ^–1(Dρ/Dt) can also be no larger than a few percent of the velocity gradients in ∇·u, such as ∂u/∂x. To see this, assume that the flow field is characterized by a length scale L, a velocity scale U, and a temperature scale δT. By this we mean that the velocity varies by U and the temperature varies by δT between locations separated by a distance of order L. The magnitude ratio of the two representative terms in the continuity equation (4.8) is:

ρ−1(u∂ρ/∂x)∂u/∂x∼(U/ρ)δρ/LU/L=δρρ=αδT≪1,

which allows (4.8) to be replaced by its incompressible form (4.10).

The Boussinesq approximation for the momentum equation is based on its form for incompressible flow, and proceeds from (4.84) divided by ρ_s:

ρρsDuDt=−1ρs∇p′+ρ′ρsg+μρs∇2u.

When the density fluctuations are small ρ/ρs≅1 and μ/ρs≅ν (= the kinematic viscosity), so this equation implies:

DuDt=−1ρ0∇p′+ρ′ρ0g+ν∇2u,

(4.86)

where ρ₀ is a constant reference value of ρ_s. This equation states that density changes are negligible when conserving momentum, except when ρ′ is multiplied by g. In flows involving buoyant convection, the magnitude of ρ′g/ρ_s is of the same order as the vertical acceleration ∂w/∂t or the viscous term ν∇²w.

The Boussinesq approximation to the energy equation starts from (4.60), written in vector notation:

ρDeDt=−p∇·u+ρε−∇·q,

(4.87)

where (4.58) has been used to insert ε, the kinetic energy dissipation rate per unit mass. Although the continuity equation is approximately ∇·u=0, an important point is that the volume expansion term p(∇·u) is not negligible compared to other dominant terms of (4.87); only for incompressible liquids is p(∇·u) negligible in (4.87). We have

−p∇·u=pρDρDt≅pρ(∂ρ∂T)pDTDt=−pαDTDt.

Assuming a perfect gas, for which p = ρRT, c_p − c_v = R, and α = 1/T, the foregoing estimate becomes:

−p∇·u=−ρRTαDTDt=−ρ(cp−cv)DTDt.

Equation (4.87) then becomes:

ρcpDTDt=ρε−∇·q,

(4.88)

where e = c_vT for a perfect gas. Note that c_v (instead of c_p) would have appeared on the left side of (4.88) if ∇·u had been dropped from (4.87).

The heating due to viscous dissipation of energy is negligible under the restrictions underlying the Boussinesq approximation. Comparing the magnitude of ρε with the left-hand side of (4.88), we obtain:

ρερcp(DT/Dt)∼2μSijSijρcpui(∂T/∂xi)∼μU2/L2ρcpU(δT/L)=νU(cpδT)L.

In typical situations this is extremely small (∼10^–7). Neglecting ρε, and assuming Fourier’s law of heat conduction (1.8) with constant k, (4.88) finally reduces to:

DTDt=κ∇2T,

(4.89)

where κ ≡ k/ρc_p is the thermal diffusivity.

Summary

The Boussinesq approximation applies if the Mach number of the flow is small, propagation of sound or shock waves is not considered, the vertical scale of the flow is not too large, and the temperature differences in the fluid are small. Then the density can be treated as a constant in both the continuity and the momentum equations, except in the gravity term. Properties of the fluid such as μ, k, and c_p are also assumed constant. Omitting Coriolis accelerations, the set of equations corresponding to the Boussinesq approximation is: (4.9) and/or (4.10), (4.86) with g = –ge_z, (4.89), and ρ = ρ₀[1 – α(T – T₀)], where the z-axis points upward. The constant ρ₀ is a reference density corresponding to a reference temperature T₀, which can be taken to be the mean temperature in the flow or the temperature at an appropriate boundary. Applications of the Boussinesq set can be found in several places in this book, for example, in the analysis of wave propagation in a density-stratified medium, thermal instability, turbulence in a stratified medium, and geophysical fluid dynamics.