Fluid entrainment

Fluid entrainment is incorporation of surrounding fluid into a moving current, jet or buoyant plume. A downstream increase in the flow's volume flux can therefore occur without an additional localized source. The added fluid generally changes the mean velocity, temperature or constituent concentration of the moving region.

For a steady jet whose surroundings initially carry little axial momentum, entrainment increases transported volume while the conserved axial momentum flux constrains how the jet slows and spreads. In a warm buoyant plume, entrained ambient air similarly dilutes the temperature anomaly while increasing the total flow. Measuring only a centreline speed cannot determine the entrainment rate; one needs integrated transport across successive sections or a suitable boundary-flux estimate. Entrainment is different from molecular diffusion, although both can contribute to mixing and are often coupled in an observed flow.

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.