Fluid entrainment

Fluid entrainment is the incorporation of surrounding fluid into a moving current, jet, or plume. It increases volume flux while diluting transported momentum, buoyancy, or concentration.

The mean speed, measured normal to a chosen plume or jet boundary, at which ambient fluid enters the turbulent region. The boundary and averaging convention must be specified: this speed is not the plume's axial velocity. In the Batchelor entrainment hypothesis for a rising Boussinesq plume, $u_e=\alpha w$, where $\alpha$ is the dimensionless entrainment coefficient and $w$ the characteristic axial speed. Multiplying by the boundary perimeter gives the entrained volume flux per axial length.

An interfacial stress $\tau =c_D{\rho }_l{u}_I^2$ supplies local stress power $P_I=S\tau u_I$. Assuming a fixed fraction $C_W$ becomes global potential energy gives ${\dot{E}}_P=C_W{P}_I$. Together with potential energy of two-layer entrainment, this gives $\dot{h}=2C_W{c}_D{u}_I^3/(g^{\prime }h)$. This is an additional local power-transfer closure. The shaft power of a rotating lid is its torque times its prescribed angular speed and need not equal $P_I$. Geometric factors and a constant transfer fraction may be absorbed into the empirical coefficients only when explicitly assumed time-independent.

In a closed tank of cross-sectional area $S$, entrainment into a uniform upper layer conserves the density deficit $({\rho }_l-{\rho }_u)h$. Thus $B=g^{\prime }h$ is constant. With depth $z$ increasing downward, gravitational potential energy is $-gS{\int }_0^H\rho (z)zdz=E_{\ast }+{\rho }_{l}SBh/2$, so its rate of increase is proportional to the rate of deepening. This includes homogenization throughout the upper layer, not merely lifting a thin parcel across the interface.

Let an initial upper mixed layer have finite depth $a$, buoyancy $b_1$, and fixed upper boundary; beneath it let $b=b_1-\Delta b+N^{2}z$ with the initial interface at $z=0$. Homogenization down to $z=-d$ conserves buoyancy and gives mixed buoyancy $b_1-[\Delta bd+N^2{d}^2/2]/(a+d)$. Integrating gravitational potential energy over the initial and final profiles gives the displayed increase per unit area. Its derivative is ${\rho }_0[\Delta ba/2+N^{2}ad/2+N^2{d}^2/4]$. Specifying absorbed mixing power is not enough to determine deepening unless the initial depth or an alternative upper-reservoir model is supplied.

In a slowly deepening mixed layer driven by a rotating boundary, a phenomenological balance between energy input, turbulent loss and acceleration of entrained fluid can keep total kinetic energy approximately fixed. For a tank of radius $R$ this gives $u^{2}h=A{\Omega }^2{R}^3$, so velocity decreases as $h^{-1/2}$. The closure does not make the speed constant and is an additional physical assumption rather than a direct consequence of mass conservation.

For potential energy of two-layer entrainment with constant $B=g^{\prime }h$, the fixed-energy turbulent mixed layer assumption gives $u_U^{2}h=C_K^2{\Omega }^2{R}^2{h}_0$. If $u_I=C_I{u}_U$ and the interfacial power model for entrainment applies with fixed coefficients, integration gives the displayed depth law with $A=5C_W{c}_D{C}_I^3{C}_K^3{\Omega }^2{R}^3/(B{h}_0)$. The relation applies only before the layer reaches the tank bottom. Its interfacial Richardson number $B{d}_I/(h{u}_I^2)$ stays constant for fixed interface thickness, whereas it decreases like $h^{-1}$ in the constant-speed entrainment model.

With constant interface thickness and the local entrainment exponent $3/2$, the fixed-energy turbulent mixed layer satisfies $\dot{h}=C{\Omega }^4{R}^6/(N^3{\delta }^{3/2}{h}^{7/2})$. Integration gives a linear growth law for $h^{9/2}$. Once an initial-depth offset is negligible, $h/R$ scales as $[{\Omega }^{2}R/(N^2\delta ){]}^{1/3}(\Omega t{)}^{2/9}$. The law applies before the mixed layer reaches the tank bottom and after the initial forcing transient.