Who is Credited With the Continuity Equation

Continuity Equation

Using the continuity equation (2.8), we can define a stream function ψ(x, zψ) where zψ is the height of the line along which ψ is constant.

From: International Geophysics , 2002

Fluid Flow Equations

John R. Fanchi , in Shared Earth Modeling, 2002

Incompressible Flow

The continuity equation for the flow of a fluid with density ρ, velocity $\overrightarrow{v}$ and no source or sink terms may be written as

(9.2.9) $\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho v\right)=0$

If we introduce the differential operator

(9.2.10) $\frac{D}{Dt}=\frac{\partial }{\partial t}+\overrightarrow{\nu }\cdot \nabla$

into Equation (9.2.9), the continuity equation has the form

(9.2.11) $\frac{D\rho }{Dt}+\rho \nabla \cdot \overrightarrow{\nu }=0$

In the case of an incompressible fluid, density is constant and the continuity equation reduces to the following condition for incompressible fluid flow:

(9.2.12) $\nabla \cdot \overrightarrow{\nu }=0$

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2D/3D Boundary Element Programming in Petroleum Engineering and Geomechanics

Morita , in Developments in Petroleum Science, 2020

9.1 Equations for fluid flow through porous media

The continuity equation and Darcy's equation for one-phase flow through porous media are given by the following forms.

(9.1) $\nabla ·\left[\mathit{\rho u}\right]=-\partial \left(\mathit{\rho \varphi }\right)/\partial t-qP\subset \Omega$

(9.2) $u=-\frac{K}{\mu }\nabla P$

The boundary conditions are specified as follows:

(9.3) $P=P_{b}P\subset {\mathrm{\Gamma }}_2$

$P=P_{i}t=0$

$q=q_{b}q\subset {\mathrm{\Gamma }}_1$

Assuming small compressibility of the fluid, the density is given by

(9.4) $\rho ={\rho }_0{e}^{c\left(P-P_0\right)}$

Hence, Eq. (9.1) is rewritten as

(9.5) $\nabla \rho ·\left[u\right]+\rho \nabla ·\left[u\right]=-\mathit{\rho c\varphi }\partial P/\partial t-qP\subset \Omega$

Unless the flowing velocity sharply changes for transient flow, the first term is relatively smaller than the second term. Then, neglecting the first term, the equation becomes

(9.6) $-\frac{\mathit{\rho K}}{\mu }{\nabla }^{2}P=-\mathit{\rho c\varphi }\partial P/\partial t-qP\subset \Omega$

Or, rewriting the equation, the following equation is derived.

(9.7) $k{\nabla }^{2}P=\mathit{c\varphi }\partial P/\partial t+q/\rho P\subset \Omega$

$\mathrm{k}=\mathrm{K}/\mathrm{\mu }$

Eq. (9.7) is an approximate equation for the single-phase fluid flow through porous media.

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Multiphase Fluid Flow Equations

John R. Fanchi , in Principles of Applied Reservoir Simulation (Fourth Edition), 2018

7.1 The Continuity Equation

The continuity equation can be derived by considering the flow of fluid into and out of a single reservoir gridblock ( Fig. 7.2). Let the symbol J denote fluid flux. Flux is defined as the rate of flow of mass per unit cross-sectional area normal to the direction of flow, which is the x direction in the present case. Assume fluid flows into the gridblock at x with flux J _x and out of the gridblock at x + Δx with flux J _x+Δx . By conservation of mass, we have the equality:

Figure 7.2. Reservoir gridblock.

$\text{mass entering the gridblock}-\text{mass leaving the gridblock}=\text{accumulation of mass in the gridblock}$

If the gridblock has length Δx, width Δy, and depth Δz, we can write the mass entering the gridblock in a time interval Δt as

(7.1) $[{(J_x)}_x\Delta y\Delta z+{(J_y)}_y\Delta x\Delta z+{(J_z)}_z\Delta x\Delta y]\Delta t=\text{Mass in}$

where we have generalized the equation to allow flux in the y and z directions as well. The notation ( J _x ) _x denotes the x direction flux at location x, with analogous meanings for the remaining terms.

Corresponding to mass entering is a term for mass exiting which has the form

(7.2) $\begin{array}{l}[{(J_x)}_{x+\Delta x}\Delta y\Delta z+{(J_y)}_{y+\Delta y}\Delta x\Delta z+{(J_z)}_{z+\Delta z}\Delta x\Delta y]\Delta t\hfill \\ +q\Delta x\Delta y\Delta z\Delta t=\text{Mass out}\hfill \end{array}$

We have added a source/sink term q which represents mass flow into (source) or out of (sink) a well. A producer is represented by q > 0, and an injector by q < 0.

Accumulation of mass in the gridblock is the change in concentration $C_{\ell }$ of phase $\ell$ in the gridblock over the time interval Δt. If the concentration $C_{\ell }$ is defined as the total mass of phase $\ell$ (oil, water, or gas) in the entire reservoir gridblock divided by the gridblock volume, then the accumulation term becomes

(7.3) $[{(C_{\ell })}_{t+\Delta t}-{(C_{\ell })}_t]\Delta x\Delta y\Delta z=\text{Mass accumulation}$

Using Eqs. (7.1) through (7.3) in the mass conservation equality

$\text{Mass in}-\text{Mass out}=\text{Mass accumulation}$

gives

(7.4) $\begin{array}{l}\left[{\left(J_x\right)}_x\mathrm{\hspace{0.17em}\Delta }y\Delta z+{\left(J_y\right)}_y\Delta x\Delta z+{\left(J_z\right)}_z\Delta x\Delta y\right]\Delta t\\ -\left[{\left(J_x\right)}_{x+\Delta x}\Delta y\Delta z+{\left(J_y\right)}_{y+\Delta y}\Delta x\Delta z+{\left(J_z\right)}_{z+\Delta z}\Delta x\Delta y\right]\Delta t\\ -q\Delta x\Delta y\Delta z\Delta t=\left[{\left(C_{\ell }\right)}_{t+\Delta t}-{\left(C_{\ell }\right)}_t\right]\Delta x\Delta y\Delta z\end{array}$

Dividing Eq. (7.4) by ΔxΔyΔzΔt and rearranging gives

(7.5) $\begin{array}{l}-\frac{{(J_x)}_{x+\Delta x}-{(J_x)}_x}{\Delta x}-\frac{{(J_y)}_{y+\Delta y}-{(J_x)}_y}{\Delta y}-\frac{{(J_z)}_{z+\Delta z}-{(J_z)}_z}{\Delta z}\hfill \\ -q=\frac{{(C_{\ell })}_{t+\Delta t}-{(C_{\ell })}_t}{\Delta t}\hfill \end{array}$

In the limit as Δx, Δy, Δz, and Δt go to zero, Eq. (7.5) becomes the continuity equation

(7.6) $-\frac{\partial J_x}{\partial x}-\frac{\partial J_y}{\partial y}-\frac{\partial J_z}{\partial z}-q=\frac{\partial C_{\ell }}{\partial t}$

The oil, water, and gas components each satisfy a mass conservation equation having the form of Eq. (7.6).

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Dynamical Paleoclimatology

In International Geophysics, 2002

9.3 The Vertically Integrated Ice-Sheet Model

The fundamental continuity equation for the rate of change of ice thickness at any point, $h_{\text{I}}\left(\lambda ,\phi \right)={\rho }_{\text{i}}^{-1}{m}_{\text{i}}$ , is Eq. (4.47), also derivable from Eq. (9.10) and the upper boundary condition, Eq. (9.17). Assuming ρ _i is a constant this equation can be written in the form

(9.28) $\frac{\partial h_{\text{I}}}{\partial t}=-\nabla \cdot {\displaystyle {\int }_{h_{\text{B}}}^h\text{v}\left(z\right)dz-\nabla \cdot h_{\text{I}}{\text{v}}_{\text{B}}+{\rho }_{\text{i}}^{-1}\left(P_{\text{I}}-M-M_{\text{B}}\right)}$

where v (z) denotes a slow ice-creep component of the total velocity v _I (z) at height z, and V _B denotes a basal sliding or "surge" component assumed to be uniform with height, to be determined by the subglacial boundary conditions (see Fig. 9-4) i.e.,

Figure 9-4. Idealized horizontal velocity components in an ice sheet, expressing the total velocity V _I (z) as the sum of an creep velocity V (z) and a basal sliding velocity V _B.

Following Paterson (1994), v _I is determined as a function of the ice thickness h _I (=h − h _B) from the equations of motion, Eqs. (9.7), (9.8), and (9.9), and the flow law, Eq. (9.12), as follows: First, integrating these equations from the top of the ice sheet (z = h), where p = 0 to any level z,

(9.30) ${\tau }_{xz}\left(z\right)=-{\rho }_{\text{i}}g\left(h-z\right)\frac{\partial h}{\partial x}$

(9.31) ${\tau }_{yz}\left(z\right)=-{\rho }_{\text{i}}g\left(h-z\right)\frac{\partial h}{\partial y}$

Then, substituting Eqs. (9.30), (9.31), and (9.32) into Eq. (9.12), and using Eq. (9.14), we obtain

(9.33) $\frac{\partial \left(u,\upsilon \right)}{\partial z}={\Xi }_{\left(x,y\right)}\cdot {\left(h-z\right)}^n$

where

${\Xi }_{\left(x,y\right)}=-2K\cdot {\left({\rho }_{\text{i}}g\right)}^n{\left[{\left(\frac{\partial h}{\partial x}\right)}^2+{\left(\frac{\partial h}{\partial y}\right)}^2\right]}^{\frac{n-1}{2}}\frac{\partial h}{\partial \left(x,y\right)}$

Integrating Eq. (9.33) from the base (z =h _B) to any level z,

(9.34) $\left(u,\upsilon \right)=\frac{{\Xi }_{\left(x,y\right)}}{n+1}\left[-{\left(h-z\right)}^{n+1}+h_{\text{I}}^{n+1}\right]+\left(u_{\text{B}},{\upsilon }_{\text{B}}\right)$

Finally, substituting Eq. (9.34) into Eq. (9.28), we obtain, in vector form, the fundamental equation governing the thickness of a column-averaged ice sheet at any point,

(9.35) $\frac{\partial h_{\text{I}}}{\partial t}=\frac{2{\left({\rho }_{\text{i}}g\right)}^n}{n+2}\nabla \cdot \left(K{h}_{\text{I}}^{n+2}{\left|\nabla h\right|}^{\frac{n-1}{2}}\right)\nabla h-\nabla \cdot h_{\text{I}}{\text{v}}_{\text{B}}+{\rho }_{\text{i}}^{-1}\left(P_{\text{i}}-M-M_{\text{B}}\right)$

This equation is the basis of many models of ice-sheet variation, some paleoclimatic applications of which are discussed in Section 9.10. The first term represents the thickness change due to the slow steady creep of ice under its own weight, and the second term represents the potentially rapid effect of the sliding velocity V_B that may include ice stream flow [cf., Eq. (9.22)].

In view of the uncertainties of basal conditions, using Eqs. (9.30) and (9.31) coupled with Eq. (9.22) in its simplest form (Weertman, 1957), with n = 1, Greve and MacAyeal (1996) and Verbitsky and Saltzman (1997) have approximated V_B for

T _B = T _M as

(9.36) ${\text{v}}_{\text{B}}=k_{\text{B}}{\underset{\_}{\tau }}_{\text{B}}=\left({\rho }_{\text{i}}g{h}_{\text{B}}\right)h_{\text{I}}\nabla h$

where k _B iS treated as a tree parameter.

It might also be possible, as suggested by Clark and Pollard (1998), to artificially represent the effects of altered states of basal water and deformable till by changing the creep coefficients n and K. A more rigorous approach than either this method or by the use of Eq. (9.36) is to deal more explicitly with the physics of subglacial water and deformable sediment (Boulton and Dobbie, 1993) starting from the governing continuity equations (see Sections 9.5 and 9.6).

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Coupled Thermo-Hydro-Mechanical-Chemical Processes in Geo-Systems

Peide Sun , in Elsevier Geo-Engineering Book Series, 2004

2.3 Solid-gas coupled model for the gas leak flow system

With the continuity equations for gas leak flow and the equilibrium equations for coal/rock mass deformation provided, the coupled mathematical model for the interaction between coal/rock mass and gas in the gas leak flow system can be described with the boundary and initial conditions ( Eq. 6) by Sun (1998).

(6) $\begin{array}{l}(T_{1t}{P}_{1.t}^2){,}_1+T_3\frac{P_1^2-P_2^2}{b_1^{·}{b}_3^{·}}=Sun(P_1,t)\frac{\partial P_1^2}{\partial t}-2P_1\frac{\partial e_1}{\partial t}\hfill \\ (T_{2t},P_{2t}^2){,}_1-T_3\frac{P_1^2-P_2^2}{b_2^{·}{b}_3^{·}}=Sun(P_2,t)\frac{\partial P_2^2}{\partial t}-2P_2\frac{\partial e_2}{\partial t}\hfill \\ U_{mj.ji}({\lambda }_m+G_m)+U_{mj.ji}{G}_m+F_{mu}+({\alpha }_m{P}_m){,}_1=0\hfill \\ e_m=U_{im,l},\hfill \\ P_m(x,y,z,0)=P_{mo}(x,y,z)\hfill \\ P_m(x_o,y_o,z_o,l){|}_{l\in (0.\leftarrow \infty )}=P_u;\hfill \\ T_{mu}\partial P_m/\partial \tilde{n}{|}_{l\in (0.\leftarrow \infty )}={\overrightarrow{Q}}_m(x,y,z,t);\hfill \\ e_m(x,y,z,t){|}_{l=0}{=}_{mv};\hfill \\ U_{mu}(x,y,z,t){|}_{t=0}=0;\hfill \\ U_m(x,y,z,t){|}_{l\in (0+\infty )}=U_{mu}.\hfill \end{array}\}$

where m = 1, 2; j = 1, 2, 3; P _m0 is the initial pore gas pressure; P _a is the pressure in free space; e _m0 is the initial deformation rate; U _mi is the boundary value for displacement; and Q _m is the quality vector for gas seepage rate from boundary coal-mass surface.

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THE DOUBLE-DECOMPOSITION CONCEPT FOR TURBULENT TRANSPORT IN POROUS MEDIA

M.J.S. DE LEMOS , in Transport Phenomena in Porous Media III, 2005

Mean flow

The development to follow assumes single-phase flow in a saturated, rigid porous medium (ΔV_f independent of time) for which, in accordance with expression (1.31), time-average operation on the variable φ commutes with the space-average. Application of the double-decomposition idea in equation (1.37) to the inertia term in the momentum equation leads to four different terms. Not all of these terms are considered in the same analysis in the literature.

Continuity

The microscopic continuity equation for an incompressible fluid flowing in a clean (non-porous) domain was given by equation (1.1) and using the double-decomposition idea of expression (1.37) gives:

(1.39) $\nabla \cdot u=\nabla \cdot \left({〈\overline{u}〉}^i+{〈u^{\prime }〉}^i+{}^i\overline{u}+{}^i{u}^{\prime }\right)=0.$

On applying both a volume- and time-average gives:

For the continuity equation, the averaging order is immaterial.

Momentum—one average operator

The transient form of the microscopic momentum equation (1.2) for a fluid with constant properties is given by the Navier-Stokes equation as follows:

(1.41) $\rho \left[\frac{\partial u}{\partial t}=\nabla \cdot (uu)\right]=–\nabla p+\mu {\nabla }^{2}u+\rho g.$

Its time-average, using $u=\overline{u}+u^{\prime }$ , gives

(1.42) $\rho \left[\frac{\partial \overline{u}}{\partial t}+\nabla \cdot (\overline{u}\overline{u})\right]=–\nabla \overline{p}+\mu {\nabla }^2\overline{u}+\nabla \cdot (–\overline{\rho u^{\prime }{u}^{\prime })}=\rho g,$

where the stresses, $–\rho \overline{u^{\prime }{u}^{\prime }}$ , are the well-known Reynolds stresses. On the other hand, the volumetric-average of equation (1.41) using the theorem of local volumetric-average, equations (1.17)–(1.19), results in the following:

(1.43) $\rho \left[\frac{\partial }{\partial t}(\varphi {〈u〉}^i)+\nabla \cdot (\varphi {〈uu〉}^i)\right]=–\nabla (\varphi {〈p〉}^i+\mu {\nabla }^2(\varphi {〈u〉}^i)+\varphi \rho g+R,$

where

(1.44) $R=\frac{\mu }{\Delta V}{\int }_{A_i}n\cdot (\nabla u)\text{d}S–\frac{1}{\Delta V}{\int }_{A_i}np\text{d}S$

represents the total drag force per unit volume due to the presence of the porous matrix, being composed by both viscous drag and form (pressure) drags. Further, using spatial decomposition to write u = 〈u〉 ⁱ + ⁱ u in the inertia term we obtain the following:

(1.45) $\begin{array}{l}\rho \left[\frac{\partial }{\partial t}\left(\varphi {〈u〉}^i\right)+\nabla \cdot \left(\varphi {〈u〉}^i{〈u〉}^i\right)\right]\\ =–\nabla (\varphi {〈p〉}^i)+\mu {\nabla }^2(\varphi {〈u〉}^i)–\nabla \cdot (\varphi {〈{}^{i}u{}^{i}u〉}^i)+\varphi \rho g+R.\end{array}$

Hsu and Cheng (1990) pointed out that the third term on the right-hand side represents the hydrodynamic dispersion due to spatial deviations. Note that equation (1.45) models typical porous media flow for Re _p ≲ 150-200. When extending the analysis to turbulent flow, time-varying quantities have to be considered.

Momentum equation—two average operators

The set of equations (1.42) and (1.45) are used when treating turbulent flow in clear fluid or low-Re _p porous media flow, respectively. In each one of those equations only one averaging operator was applied, either time or volume, respectively. In this work, an investigation on the use of both operators in now conducted with the objective of modeling turbulent flow in porous media.

The volume-average of equation (1.42) gives for the time-mean flow in a porous medium:

(1.46) $\begin{array}{l}\rho \left[\frac{\partial }{\partial t}(\varphi {〈\overline{u}〉}^i)+\nabla \cdot \left(\varphi {〈\overline{u}\overline{u}〉}^i)\right)\right]\\ =–\nabla \cdot (\varphi {〈\overline{p}〉}^i)+\mu {\nabla }^2(\varphi {〈\overline{u}〉}^i)+\nabla \cdot \left(–\rho \varphi {〈\overline{u^{\prime }{u}^{\prime }}〉}^i\right)+\varphi \rho g+\overline{R},\end{array}$

where

(1.47) $\overline{R}=\frac{\mu }{\nabla V}{\int }_{A_i}n\cdot (\nabla \overline{u})\text{d}S–\frac{1}{\nabla V}{\int }_{A_i}n\overline{p}\text{d}S$

is the time-averaged total drag force per unit volume ('body force'), due to solid particles, composed by both viscous and form (pressure) drags.

Likewise, applying now the time-average operation to equation (1.43), we obtain:

(1.48) $\begin{array}{l}\rho \left[\frac{\partial }{\partial t}\overline{(\varphi {〈\overline{u}+u^{\prime }〉}^i)}+\nabla \cdot \overline{(\varphi {〈\left(\overline{u}+u^{\prime }\right)(\overline{u}+u^{\prime })〉}^i)}\right]\\ =–\nabla \overline{(\varphi {〈\overline{p}+p^{\prime }〉}^i)}+\mu {\nabla }^2\overline{(\varphi {〈\overline{u}+u^{\prime }〉}^i)}+\varphi \rho g+\overline{R}.\end{array}$

Dropping terms containing only one fluctuating quantity results in:

(1.49) $\begin{array}{l}\rho \left[\frac{\partial }{\partial t}\left(\varphi {〈\overline{u}〉}^i\right)+\nabla \cdot (\varphi {〈\overline{u}\overline{u}〉}^i)\right]\\ =–\nabla (\varphi {〈\overline{p}〉}^i)+\mu {\nabla }^2(\varphi {〈\overline{u}〉}^i)+\nabla \cdot \left(–\rho \varphi {〈\overline{u^{\prime }{u}^{\prime }}〉}^i\right)+\varphi \rho g+\overline{R},\end{array}$

where

(1.50) $\begin{array}{lll}\overline{R}& =& \frac{\mu }{\Delta V}{\int }_{A_i}n\cdot \left[\nabla \overline{(\overline{u}+u^{\prime })}\right]\text{d}S–\frac{1}{\Delta V}{\int }_{A_i}n\overline{(\overline{p}+p^{\prime })}\text{d}S\\ & =& \frac{\mu }{\Delta V}{\int }_{A^i}n\cdot (\nabla \overline{u})\text{d}S–\frac{1}{\Delta V}{\int }_{A_i}n\overline{p}\text{d}S.\hfill \end{array}$

Comparing equations (1.46) and (1.49), we can see that also for the momentum equation the order of the application of both averaging operators is immaterial.

It is interesting to emphasize that both views in the literature use the same final form for the momentum equation. The term $\overline{\text{R}}$ is modeled by the Darcy-Forchheimer (Dupuit) expression after either order of application of the average operators. Since both orders of integration lead to the same equation, namely expression (1.47) or (1.50), there would be no reason for modeling them in a different form. Had the outcome of both integration processes been distinct, the use of a different model for each case would have been consistent. In fact, it has been pointed out by Pedras and de Lemos (2000b) that the major difference between those two paths lies in the definition of a suitable turbulent kinetic energy for the flow. Accordingly, the source of controversies comes from the inertia term, as seen below.

Inertia term—space and time (double) decomposition

Applying the double-decomposition idea seen before for velocity (equation (1.37)) to the inertia term of equation (1.41) will lead to different sets of terms. In the literature, not all of them are used in the same analysis.

Starting with time decomposition and applying both average operators, see equation (1.46), gives:

(1.51) $\nabla \cdot \overline{(\varphi {〈uu〉}^i)}=\nabla \cdot \overline{\left(\varphi {〈(\overline{u}+u^{\prime })(\overline{u}+u^{\prime })〉}^i\right)}=\nabla \cdot \left[\varphi \left({〈\overline{u}\overline{u}〉}^i+{〈\overline{u^{\prime }{u}^{\prime }}〉}^i\right)\right].$

Using spatial decomposition to write $\overline{u}={〈\overline{u}〉}^i{+}^i\overline{u}$ we obtain:

(1.52) $\begin{array}{l}\nabla \cdot \left[\varphi \left({〈\overline{u}\overline{u}〉}^i+{〈\overline{u^{\prime }{u}^{\prime }}〉}^i\right)\right]=\nabla \cdot \left\{\varphi \left[{〈\left({〈\overline{u}〉}^i+{}^i\overline{u}\right)\left({〈\overline{u}〉}^i+{}^i\overline{u}\right)〉}^i+{〈\overline{u^{\prime }{u}^{\prime }}〉}^i\right]\right\}\\ =\nabla \cdot \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+{〈{}^i\overline{u}{}^i\overline{u}〉}^i+{〈\overline{u^{\prime }{u}^{\prime }}〉}^i\right]\right\}\end{array}$

Now, applying equation (1.30) to write u ′ = 〈 u ′〉 ⁱ + ⁱ u ′, and substituting into expression (1.52) gives:

(1.53) $\begin{array}{l}\nabla \cdot \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+{〈{}^i\overline{u}{}^i\overline{u}〉}^i+{〈\overline{u^{\prime }{u}^{\prime }}〉}^i\right]\right\}\\ =\nabla \cdot \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+{〈{}^i\overline{u}{}^i\overline{u}〉}^i+{〈\overline{\left({〈u^{\prime }〉}^i+{}^i{u}^{\prime }\right)\left({〈u^{\prime }〉}^i+{}^i{u}^{\prime }\right)}〉}^i\right]\right\}\\ =\nabla \cdot \{\varphi [{〈\overline{u}〉}^i{〈\overline{u}〉}^i+{〈{}^i\overline{u}{}^i\overline{u}〉}^i\\ +{〈\overline{{〈u^{\prime }〉}^i{〈u^{\prime }〉}^i+{〈u^{\prime }〉}^i{}^i{u}^{\prime }+{}^i{u}^{\prime }{〈u^{\prime }〉}^i+{}^i{u}^{\prime }{}^i{u}^{\prime }}〉}^i]\}\\ =\nabla \cdot \{\varphi [{〈\overline{u}〉}^i{〈\overline{u}〉}^i+{〈{}^i\overline{u}{}^i\overline{u}〉}^i+\overline{〈u^{\prime }〉{}^i〈u^{\prime }〉{}^i}\\ \overline{{〈{〈u^{\prime }〉}^i{}^i{u}^{\prime }〉}^i}+\overline{{〈{}^i{u}^{\prime }{〈u^{\prime }〉}^i〉}^i}+\overline{{〈{}^i{u^{\prime }}^i{u}^{\prime }〉}^i}]\}.\end{array}$

The fourth and fifth terms on the right-hand side contains only one space-varying quantity and will vanish under the application of volume integration. Equation (1.53) will then be reduced to

(1.54) $\nabla \cdot \overline{\left(\varphi {〈uu〉}^i\right)}=\nabla \cdot \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+\overline{{〈u^{\prime }〉}^i{〈u^{\prime }〉}^i}+{〈{}^i\overline{u}{}^i\overline{u}〉}^i+{\overline{〈{}^i{u}^{\prime }{}^i{u}^{\prime }〉}}^i\right]\right\}$

Using the equivalence (1.31) and (1.32), equation (1.54) can be further rewritten as follows:

(1.55) $\nabla \cdot \overline{\left(\varphi {〈uu〉}^i\right)}=\nabla \cdot \left\{\varphi \left[\overline{{〈\overline{u}〉}^i{〈\overline{u}〉}^i}+\overline{{〈u^{\prime }〉}^i{〈u〉}^{i^{\prime }}}+{〈\overline{{}^{i}u{}^{i}u}〉}^i+{\overline{〈{}^i{u}^{\prime }{}^i{u}^{\prime }〉}}^i\right]\right\}$

with an interpretation of the terms in equation (1.54) given later.

Another route to follow to reach the same results is to start out with the application of the space decomposition in the inertia term, as usually done in classical mathematical treatment of porous media flow analysis. Then we obtain

(1.56) $\begin{array}{lll}\nabla \cdot \overline{(\varphi {〈uu〉}^i)}& =& \nabla \cdot [\varphi \overline{{〈\left({〈u〉}^i+{}^{i}u\right)\left({〈u〉}^i+{}^{i}u\right)〉}^i]}\\ & =& \nabla \cdot \overline{\left[\varphi \left({〈u〉}^i{〈u〉}^i+{〈{}^{i}u{}^{i}u〉}^i\right)\right],}\end{array}$

and on time-averaging the right-hand side, using equation (1.33) to express ${〈u〉}^i={〈\overline{u}〉}^i+{〈u^{\prime }〉}^i$ , this becomes:

(1.57) $\begin{array}{lll}\nabla \cdot \overline{\left[\varphi \left({〈u〉}^i{〈u〉}^i+{〈{}^{i}u{}^{i}u〉}^i\right)\right]}& =& \nabla \cdot \left\{\overline{\varphi \left[\left({〈\overline{u}〉}^i+{〈u^{\prime }〉}^i\right)\left({〈\overline{u}〉}^i+{〈u^{\prime }〉}^i\right)+{〈{}^{i}u{}^{i}u〉}^i\right]}\right\}\\ & =& \nabla \cdot \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+\overline{{〈u^{\prime }〉}^i{〈u^{\prime }〉}^i}+\overline{{〈{}^{i}u{}^{i}u〉}^i}\right]\right\}.\end{array}$

With the help of equation (1.34) one can write ⁱ u = ⁱ ū + ⁱ u ′ which, inserted into expression (1.57), gives:

(1.58) $\begin{array}{lll}\nabla & \cdot & \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+\overline{{〈u^{\prime }〉}^i{〈u^{\prime }〉}^i}+\overline{{〈{}^{i}u{}^{i}u〉}^i}\right]\right\}\\ & =& \nabla \cdot \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+{\overline{{〈u^{\prime }〉}^i〈u^{\prime }〉}}^i+\overline{{〈({}^i\overline{u}+{}^i{u}^{\prime })({}^i\overline{u}+{}^i{u}^{\prime })〉}^i}\right]\right\}\hfill \\ & =& \nabla \cdot \left\{\varphi \left[{〈\overline{u}〉}^i{〈\overline{u}〉}^i+\overline{{〈u^{\prime }〉}^i{〈u^{\prime }〉}^i}+{〈\overline{{}^i\overline{u}{}^i\overline{u}}+\overline{{}^i\overline{u}{}^i{u}^{\prime }}+\overline{{}^i{u}^{\prime }{}^i\overline{u}}+\overline{{}^i{u}^{\prime }{}^i{u}^{\prime }}〉}^i\right]\right\}.\hfill \end{array}$

Application of the time-average operator to the fourth and fifth terms on the right-hand side of equation (1.58), containing only one fluctuating component, vanishes it. In addition, remembering that with expression (1.32) the equivalences ⁱ ū = $\overline{{}^{i}u}$ and 〈 u ′〉 ⁱ = 〈 u 〉 ^i′ are valid, and that with expression (1.31) we can write $\overline{〈u〉{}^i}$ = 〈 ū 〉 ⁱ , we obtain the following alternative form for equation (1.58):

(1.59) $\begin{array}{l}\nabla \cdot \overline{\left[\varphi \left({〈u〉}^i{〈u〉}^i+{〈{}^{i}u{}^{i}u〉}^i\right)\right]}\\ =\nabla \cdot \left\{\varphi \left({〈\overline{u}〉}^i{〈\overline{u}〉}^i+\overline{{〈u^{\prime }〉}^i{〈u^{\prime }〉}^i}+{〈{}^i\overline{u}{}^i\overline{u}〉}^i+{〈\overline{{}^i{u}^{\prime }{}^i{u}^{\prime }}〉}^i\right)\right\},\\ \begin{array}{c}\uparrow \\ \text{I}\end{array}\begin{array}{c}\uparrow \\ \text{II}\end{array}\begin{array}{c}\uparrow \\ \text{III}\end{array}\begin{array}{c}\uparrow \\ \text{IV}\end{array}\end{array}$

which is the same result as expression (1.54).

The physical significance of all four terms on the right-hand side of (1.59) can be discussed as follows.

I

Convective term of macroscopic mean velocity.

II

Turbulent (Reynolds) stresses divided by the density ρ due to the fluctuating component of the macroscopic velocity.

III

Dispersion associated with spatial fluctuations of microscopic time-mean velocity. Note that this term is also present in the laminar flow, or say, when Re _p < 150.

IV

Turbulent dispersion in a porous medium due to both time and spatial fluctuations of the microscopic velocity.

Further, the macroscopic Reynolds stress tensor (MRST) is given in Pedras and de Lemos (2001a) based on equation (1.25) as follows:

(1.60) $–\rho \varphi {〈\overline{u^{\prime }{u}^{\prime }}〉}^i={\mu }_{t_{\varphi }}2{〈\overline{D}〉}^v–\frac{2}{3}\varphi \rho {〈k〉}^{i}I,$

where

(1.61) ${〈\overline{D}〉}^v=\frac{1}{2}\left\{\nabla (\varphi {〈\overline{u}〉}^i+{[\nabla (\varphi {〈\overline{u}〉}^i)]}^{\text{T}}\right\}$

is the macroscopic deformation tensor, 〈k〉 ⁱ is the intrinsic average for k, and μ _{t ϕ} is the macroscopic turbulent viscosity assumed to be in Pedras and de Lemos (2001c) as follows:

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NUMERICAL MODELS | Chemistry Models

M.P. Chipperfield , S.R. Arnold , in Encyclopedia of Atmospheric Sciences (Second Edition), 2015

Chemical Families

The number of continuity equations to be solved (and computational time) can be reduced by grouping closely coupled chemical species together in a family. As well as needing to solve only one continuity equation, the photochemical lifetime of the family is generally longer than the lifetimes of the individual members, producing a less stiff system ( Figure 2). Finally, using chemical families has advantages in multidimensional models. Generally, it is not desirable to transport short-lived species separately as they have strong gradients (e.g., near the terminator), which can cause numerical problems (undershoots and overshoots) in advection schemes. A chemical family will generally have a smoother distribution and pose fewer problems for the advection scheme.

Figure 2. Photochemical lifetimes (defined as 1/(first-order loss rate)) of Cl, ClO, and ClOx (= Cl + ClO). The ClOx family has a much longer lifetime than the shortest lived family member, resulting in a less stiff system of equations to solve.

In stratospheric models a ClOx family is often defined as [ClOx] = [ClO] + [Cl]. This is justified because Cl is in rapid photochemical equilibrium with ClO, and change in the concentration of ClO will also affect Cl through the reactions, which interconvert the two. When a chemical family is used in a model, a single chemical continuity equation is written for the overall rate of change of the family. Based on the reactions given in Table 1, the continuity equation for ClOx can be expressed by eqn [7], where M represents any air molecule.

[7] $\begin{array}{l}\frac{\text{d}\left[\mathrm{ClOx}\right]}{\text{d}t}=0\\ =-k_{\mathrm{VI}}\left[\mathrm{ClO}\right]\left[\mathrm{N}{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+J_{\mathrm{VII}}\left[\mathrm{ClON}{\mathrm{O}}_2\right]\end{array}$

Note that reaction [III] for example, which simply interconverts Cl and ClO has no net effect of ClOx and does not appear in eqn [7]. The concentration of the total family must be divided among the n individual members. This is achieved by writing n − 1 steady state expressions for n − 1 members. In the case of the ClOx family, by placing Cl in steady state (d[ClO]/dt = 0) we can derive eqn [8] for the ratio of [Cl]/[ClO].

[8] $\frac{\left[\mathrm{Cl}\right]}{\left[\mathrm{ClO}\right]}=\frac{k_{\mathrm{IV}}\left[\mathrm{O}\right]+k_{\mathrm{V}}\left[\mathrm{N}\mathrm{O}\right]+J_{\mathrm{VII}}\frac{\left[\mathrm{ClON}{\mathrm{O}}_2\right]}{\left[\mathrm{ClO}\right]}}{k_{\mathrm{III}}\left[{\mathrm{O}}_3\right]}$

Although this equation is derived by assuming Cl is in steady state, the concentration of Cl (and ClO) will vary over the model time step as ClOx changes. However, eqn [7] effectively fixes the ratio of Cl:ClO over this time step.

Care is needed when deriving these expressions for the partitioning of family members. Most of the terms in eqn [7] can be identified with reactions [III], [IV], and [V], which directly interconvert Cl and ClO. However, there is also a term involving [ClONO₂]/[ClO], which is related to the two-step interconversion of ClO and Cl via the formation and photolysis of ClONO₂. It is very important to include these indirect terms as they are often associated with catalytic cycles that destroy stratospheric O₃ via the reaction [III]. In order for the model to correctly determine the O₃ loss, the calculated [Cl] must be accurate.

Another chemical family commonly used in atmospheric models is 'odd oxygen,' which is defined as Ox = O(³P) + O(¹D) + O₃. This family provides a very convenient way of calculating the atmospheric abundances of O₃, O(³P), and O(¹D) below about 70 km. Above this altitude the photochemical lifetime of O becomes long (due to the low air density) and so, O and O₃ can no longer be assumed to be in photochemical equilibrium.

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Water and the atmosphere

Daniel A. Vallero , in Air Pollution Calculations, 2019

7.6 Intercompartmental exchange to and from the hydrosphere

The total amount of water in the hydrosphere is constant, but its location and quality vary in time and space. Air pollution managers and engineers must account for possible changes in weather, climate, and other hydrologic conditions with time. It is reasonable and prudent to expect changes over the life of a design. Mean global and local temperature increases due to increased atmospheric concentrations of global greenhouse should be factored into designs and facility citing decisions [13, 14]. For example, certain facilities should be hardened or simply not be built in areas potentially affected by rising water levels on coasts and on other large water bodies, for example, the Great Lakes in North America. In addition, the relationships between water supplies and engineered systems can be affected by changes in hydrologic and hydraulic conditions, for example, the saltwater wedge encroaching inward (see Fig. 7.7).

Fig. 7.7. Saltwater intrusion into a freshwater system. This denser salt water submerges under the lighter freshwater system. The same phenomenon can occur in coastal aquifers.

The density difference between fresh and salt water can be a slowly unfolding disaster for the health of people living in coastal communities and for marine and estuarine ecosystems. Salt water contains a significantly greater mass of ions than does fresh water (see Table 7.7). The denser saline water can wedge beneath fresh waters and pollute surface waters and groundwater. This phenomenon, known as saltwater intrusion, can significantly alter an ecosystem's structure and function and threaten freshwater organisms. The literature often draws abrupt distinctions between freshwater and saltwater environments, but in fact, there are large transition zones between the two. For example, an estuary is a partially enclosed coastal water body that connects a freshwater system, for example, a river, to water that is influenced by tides and within which there is a gradient between salt water and fresh water.

Table 7.7. Important general ionic composition classifications of fresh waters and marine waters

Composition	River water	Salt water
pH	6–8	8
Ca2   +	4   ×   10−   5M	1   ×   10−   2M
Cl−	2   ×   10−   4M	6   ×   10−   1M
HCO3 −	1   ×   10−   4M	2   ×   10−   3M
K+	6   ×   10−   5M	1   ×   10−   2M
Mg2   +	2   ×   10−   4M	5   ×   10−   2M
Na+	4   ×   10−   4M	5   ×   10−   1M
SO4 2   −	1   ×   10−   4M	3   ×   10−   2M

M   =  molarity (moles per liter solution).

Sources of data: P.M. Gschwend, Environmental Organic Chemistry. John Wiley &amp; Sons, 2016; J.P. Kim, K.A. Hunter, M.R. Reid, Factors influencing the inorganic speciation of trace metal cations in fresh waters, Mar. Freshw. Res. 50 (4) (1999) 367–372; R.P. Schwarzenbach, P.M. Gschwend, D.M. Imboden, Organic acids and bases: acidity constant and partitioning behavior, in: Environmental Organic Chemistry, 1993, pp. 245–274; D. Vallero, Environmental Biotechnology: A Biosystems Approach, Elsevier Science, 2015.

The effect in sensitive coastal habitats like those in South Florida would be exacerbated by the rising sea level, which could submerge low-lying areas of the Everglades (see Fig. 7.8), increasing the salinity in portions of the aquifer. The rising seawater could force salty waters upstream into coastal areas, thus threatening surface water supplies. Furthermore, siting pollution control technologies in these sensitive areas would be sources of additional contaminants, including air pollutants. Similar problems could also occur in northeastern US aquifers that are recharged by fresh portions of streams that are vulnerable to increased salinity during severe droughts. [13] Indeed, similar groundwater-surface water-seawater-atmospheric interactions could be affected across the globe. Planning, engineering, and construction in these areas require methods and tools to optimize siting alternatives, including geographic information systems (GIS) [15].

Fig. 7.8. Elevation and aquifer locations in southern Florida. Although a small part of the aquifer is beneath salty mangrove area, most of it is recharged by the freshwater Everglades, rendering the area vulnerable to saltwater intrusion and increased salinity of both surface and groundwater sources of drinking water.

(Source: U.S. Environmental Protection Agency. Saving Florida's Vanishing Shores. (2002). 1 November 2013. Available: http://www.epa.gov/climatechange/Downloads/impacts-adaptation/saving_FL.pdf.)

Salinity of water is a relative term. The values in Table 7.7 are at best averages and target concentrations. Note that the classifications using two dominant ions, Na⁺ and Cl⁻, differ by three orders of magnitude between "fresh" and "saline" waters. The same is true for bromine (Br), fluorine (F), boron (B), magnesium (Mg), and calcium (Ca). Consider the hypothetical example in Fig. 7.9 of these ionic strengths before and after saltwater intrusion. Although the water near the water supply is not "salt water" nor does it currently violate the secondary drinking water standard, the ionic concentrations are cause for concern and serve as a warning that the trend is likely to be toward even higher salinity.

Fig. 7.9. Hypothetical isopleths of total dissolved solid (mg   L^{−   1}) concentrations in groundwater (e.g., 30   m depth). The increases over the decade, that is, increased concentrations in 2013 (A) versus 2003 (B), indicate saltwater intrusion and potential contamination at the drinking water well site.

(Source: D. Vallero, Environmental Biotechnology: A Biosystems Approach, Elsevier Science, 2015.)

Salinity is the total dissolved solids (TDS), not just ionic composition. As mentioned, the dissolved solids can be several orders of magnitude higher in salt water than fresh water, but this is not always true for the nutrients. It is true for potassium (K), which has a mean concentration of 2.3   mg   L^{−   1} in fresh water but 416   mg   L^{−   1} in seawater with 35% salinity. However, nitrogen (N) and phosphorus (P) do not differ substantially, with N having 0.25   mg   L^{−   1} in fresh water and 0.5   mg   L^{−   1} in seawater and P having 0.02   mg   L^{−   1} in fresh water and 0.07   mg   L^{−   1} in seawater [16].

The foregoing discussion illustrates how an indirect effect like the warming of the atmosphere from increasing concentrations of greenhouse gases can lead to hydrospheric impacts, for example, rising sea level and water contamination. The interconnectedness of the atmosphere, hydrosphere, and biosphere is complex and extensive (see Fig. 7.10). A small change in one small part of the spheres can lead to unanticipated outcomes. Thus, hydrologic cycling specifically and biogeochemical cycling generally are sensitive to initial conditions and, as such, must be treated as chaotic systems.

Fig. 7.10. Potential impacts of climate change of the hydrologic cycle.

(Source: U.S. Environmental Protection Agency. Climate Impacts on Water Resources. (2016). 1 March 2018. Available: https://19january2017snapshot.epa.gov/climate-impacts/climate-impacts-water-resources_.html. Based on information from T.R. Karl, J. Melillo, T. Peterson, Global climate change impacts in the United States, Global Climate Change Impacts in the United States, 2009.)

The basic hydrologic continuity equation states that a water system storage changes at a rate of the difference between water entering the system ( I) and the amount of water exiting the system (O):

(7.35) $\frac{\mathit{dS}}{\mathit{dx}}=I-O$

During the month of July, an aquifer is being pumped to irrigate a corn crop at 20   m³  sec^{−   1} and is being recharged with rainwater at a net rate of 5   m³  sec^{−   1}; what is the change in volume during July?

Solving for dS:

$\mathit{dS}=\left(I-O\right)\mathit{dt}$

$\mathit{dS}=\left(5–20\right){\mathrm{m}}^3{sec}^{-1}\times \left(31\text{days}\times 24\mathrm{h}{\mathrm{day}}^{-1}\times 60min\mathrm{hr}.{}^{-1}\times 60\right)sec$

$\mathit{dS}\approx -4\times {10}^7{\mathrm{m}}^3$

Thus, the aquifer is losing 40 million cubic meters of water (about ten and a half billion gallons) in July.

The continuity equation can be further described as.

(7.36) $\frac{\mathit{dS}}{\mathit{dx}}=P-R-E-T-I$

where P is precipitation, R is surface runoff (e.g., to streams), E is evaporation, T is transpiration, and I is infiltration (i.e., the movement of water through the troposphere-soil interface).

During August and September, Townsville (area (A)   =   50   km ² ) received 250   mm of rain. The evaporation rate was 100   mm, transpiration was 50   mm, and infiltration was 30   mm. What was the volume of surface runoff during this period?

Solving for R:

$R=P-E-T-I$

$R=\left(250–100–50–30\right)\mathrm{mm}$

$R=70\mathrm{mm}=0.07\mathrm{m}$

Volume of surface runoff (V) is R  × A, so

$V=0.07\mathrm{m}\times 50\times {10}^6{\mathrm{m}}^2$

$V=3.5\times {10}^6{\mathrm{m}}^2$

Nearby city, Villetown, is also 50   km ² in area. They have instituted a number of drainage improvements, including tree plantings and permeable surfaces in parking lots. During the same August and September, Villetown also received 250   mm of rain. However, the evaporation rate was 60   mm, transpiration was 60   mm, and infiltration was 100   mm. What was the volume of surface runoff for Villetown for this time period?

Solving for R:

$R=P-E-T-I$

$R=\left(250–60–60–100\right)\mathrm{mm}$

$R=30\mathrm{mm}=0.03\mathrm{m}$

Volume of surface runoff (V) is R  × A, so.

$V=0.03\mathrm{m}\times 50\times {10}^6{\mathrm{m}}^2$

$V=1.5\times {10}^6{\mathrm{m}}^2$

Therefore, Villetown's surface runoff was 2 million cubic meters less than Townville's during this 2-month period, in part at least to the increased infiltration and decreased evaporation rates. The transpiration rate increased due to the greater number of trees. Thus, the tree planting increased E, T, and I, by holding water in the plant tissue, releasing water to the atmosphere, and moving water to the soil via the root systems.

Note that all the variables on the right side of Eq. (7.10) are fluxes, adhering to the diffusion, advective, and turbulence principles discussed in Chapter 6.

The mass of water vapor and air is of the same units, usually kilograms. If water vapor in the parcel has the same temperature and pressure as the air, then.

(7.37) $\omega =0.62198\bullet \frac{p_w}{p_{\mathit{at}}-p_w}$

where p _w is the partial pressure of the water vapor in the moist air parcel and p _at is the atmospheric pressure.

The degree of saturation (μ) is the ratio of ω of moist air to humidity ratio of saturated moist air (ω _sat ) at a specific temperature and pressure:

(7.38) $\mu =\frac{\omega }{{\omega }_{\mathit{sat}}}$

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Applications of the momentum principle: hydraulic jump, surge and flow resistance in open channels

Hubert Chanson , in Hydraulics of Open Channel Flow (Second Edition), 2004

4.2.2 Basic equations

For a steady flow in a horizontal rectangular channel of constant channel width, the three fundamental equations become:

(A)

Continuity equation

(4.2) $Q=V_1{d}_{1}B=V_2{d}_{2}B$

where: V ₁ and d ₁ are, respectively, the velocity and flow depth at the upstream cross-section (Fig. 4.4), V ₂ and d ₂ are defined at the downstream cross-section, B is the channel width and Q is the total flow rate.

Fig. 4.4. Application of the momentum equation to a hydraulic jump.

(B)

Momentum equation (Bélanger equation) ¹ forces. The momentum equation states that the sum of all the forces acting on the control volume equals the change in momentum flux across the control volume. For a hydraulic jump, it yields:

(4.3) $\left(\frac{1}{2}\rho g{d}_1^2-\frac{1}{2}\rho g{d}_2^2\right)B-F_{\text{fric}}=\rho Q(V_2-V_1)$

where F _fric is a drag force exerted by the channel roughness on the flow (Fig. 4.4).

(C)

Energy equationThe energy equation (2.24) can be transformed as:

(4.4a) $H_1=H_2+\Delta H$

where ΔH is the energy loss (or head loss) at the jump, and H ₁ and H ₂ are upstream and downstream total heads, respectively. Assuming a hydrostatic pressure distribution and taking the channel bed as the datum, equation (4.4a) becomes:

(4.4b) $\frac{V_1^2}{2g}+d_1=\frac{V_2^2}{2g}+d_2+\Delta H$

Note that equations (4.1) – (4.4) were developed assuming hydrostatic pressure distributions at both the upstream and downstream ends of the control volume (Fig. 4.4). Furthermore, the upstream and downstream velocity distributions were assumed uniform for simplicity.

Notes

1.

In the momentum equation, (ρV) is the momentum per unit volume.

2.

In the simple case of uniform velocity distribution, the term (ρVV) is the momentum flux per unit area across the control surface. (ρVVA) is the total momentum flux across the control surface.

Neglecting the drag force on the fluid, the continuity and momentum equations provide a relationship between the upstream and downstream flow depths as:

(4.5a) $d_2=\sqrt{{\left(\frac{d_1}{2}\right)}^2+\frac{2Q^2}{g{d}_1{B}^2}}-\frac{d_1}{2}$

or in dimensionless terms:

(4.5b) $\frac{d_2}{d_1}=\frac{1}{2}\left(\sqrt{1+8F{r}_1^2}-1\right)$

where Fr ₁ is the upstream Froude number: $F{r}_1={\text{V}}_1/\sqrt{g{d}_1}.$ It must be noted that Fr ₁ > 1. The depths d ₁ and d ₂ are referred to as conjugate depths (or sequent depths). Using equation (4.5) the momentum equation yields:

(4.6) $F{r}_2=\frac{2^{3/2}F{r}_1}{{\left(\sqrt{1+8F{r}_1^2}-1\right)}^{3/2}}$

where Fr ₂ is the downstream Froude number. The energy equation gives the head loss:

(4.7a) $\Delta H=\frac{{\left(d_2-d_1\right)}^3}{4d_1{d}_2}$

and in dimensionless terms:

(4.7b) $\frac{\Delta H}{d_1}=\frac{{\left(\sqrt{1+8F{r}_1^2}-3\right)}^3}{16\left(\sqrt{1+8F{r}_1^2}-1\right)}$

Equations (4.5) – (4.7) are summarized in Fig. 4.5. Figure 4.5 provides means to estimate rapidly the jump properties as functions of the upstream Froude number. For example, for Fr ₁ = 5, we can deduce Fr ₂ ~ 0.3, d ₂/d ₁ ~ 6.5, ΔH/d ₁ ~ 7.

Fig. 4.5. Flow properties downstream of a hydraulic jump in a rectangular horizontal channel.

Notes

1.

In a hydraulic jump, the downstream flow depth d ₂ is always larger than the upstream flow depth d ₁.

2.

The main parameter of a hydraulic jump is its upstream Froude number Fr ₁.

3.

If only the downstream flow conditions are known, the solution of the continuity and momentum equations is given by:

$\frac{d_1}{d_2}=\frac{1}{2}\left(\sqrt{1+8F{r}_2^2}-1\right)$

4.

A hydraulic jump is a very effective way of dissipating energy. It is also an effective mixing device. Hydraulic jumps are commonly used at the end of spillway or in dissipation basin to destroy much of the kinetic energy of the flow. The hydraulic power dissipated in a jump equals; ρgQΔH where ΔH is computed using equation (4.7).

5.

Hydraulic jumps are characterized by air entrainment. Air is entrapped at the impingement point of the supercritical inflow with the roller. The rate of air entrainment may be approximated as:

$\begin{array}{cc}\frac{Q_{\text{air}}}{Q}\approx 0.018{(F{r}_1-1)}^{1.245}& \text{(Rajaratnam,1967)}\end{array}$

$\begin{array}{cc}\frac{Q_{\text{air}}}{Q}\approx 0.014{(F{r}_1-1)}^{1.4}& \text{(Wisner,1965)}\end{array}$

Wood (1991) and Chanson (1997) presented complete reviews of these formulae.

If the jump is in a closed duct then, as the air is released, it could accumulate on the roof of the duct. This phenomena is called 'blowback' and has caused failures in some cases (Falvey, 1980).

6.

Recent studies showed that the flow properties (including air entrainment) of hydraulic jumps are not only functions of the upstream Froude number but also of the upstream flow conditions: i.e. partially developed or fully developed inflow. The topic is currently under investigation.

Application

Considering a hydraulic jump in a horizontal rectangular channel, write the continuity equation and momentum principle. Neglecting the boundary shear, deduce the relationships d ₂/d ₁ = f(Fr ₁) and Fr ₂ = f(Fr ₁).

Solution

The continuity equation and the momentum equation (in the flow direction) are respectively:

[C] $q=V_1{d}_1=V_2{d}_2$

[M] $\frac{1}{2}\rho g{d}_1^2-\frac{1}{2}\rho g{d}_2^2=\rho q(V_2-V_1)$

where q is the discharge per unit width.

[C] implies V ₂ = V ₁ d ₁/d ₂. Replacing [C] into [M], it yields:

$\frac{1}{2}\rho g{d}_1^2-\frac{1}{2}\rho g{d}_2^2=\rho V_1^2{d}_1\left(\frac{d_1}{d_2}\right)-\rho V_1^2{d}_1$

Dividing by $(\rho g{d}_1^2)$ it becomes:

$\frac{1}{2}-\frac{1}{2}{\left(\frac{d_2}{d_1}\right)}^2=F{r}_1^2\left(\frac{d_1}{d_2}\right)-F{r}_1^2$

After transformation we obtain a polynomial equation of degree three in terms of d ₂/d ₁:

$\frac{1}{2}{\left(\frac{d_2}{d_1}\right)}^3-\left(\frac{1}{2}+F{r}_1^2\right)\left(\frac{d_2}{d_1}\right)+F{r}_1^2=0$

or

$\frac{1}{2}\left(\frac{d_2}{d_1}-1\right)\left({\left(\frac{d_2}{d_1}\right)}^2+\left(\frac{d_2}{d_1}\right)-2F{r}_1^2\right)=0$

The solutions of the momentum equation are the obvious solution d ₂ = d ₁ and the solutions of the polynomial equation of degree two:

${\left(\frac{d_2}{d_1}\right)}^2+\left(\frac{d_2}{d_1}\right)-2F{r}_1^2=0$

The (only) meaningful solution is:

$\frac{d_2}{d_1}=\frac{1}{2}\left(\sqrt{1+8F{r}_1^2}-1\right)$

Using the continuity equation in the form V ₂ = V ₁ d ₁/d ₂, and dividing by $\sqrt{g{d}_2},$ it yields:

$F{r}_2=\frac{V_2}{\sqrt{g{d}_2}}=\frac{V_1}{\sqrt{g{d}_1}}{\left(\frac{d_1}{d_2}\right)}^{3/2}=F{r}_1\frac{2^{3/2}}{{\left(\sqrt{1+8F{r}_1^2}-1\right)}^{3/2}}$

Application

A hydraulic jump takes place in a 0.4 m wide laboratory channel. The upstream flow depth is 20 mm and the total flow rate is 31 l/s. The channel is horizontal, rectangular and smooth. Calculate the downstream flow properties and the energy dissipated in the jump. If the dissipated power could be transformed into electricity, how many 100 W bulbs could be lighted with the jump?

Solution

The upstream flow velocity is deduced from the continuity equation:

$V_1=\frac{Q}{B{d}_1}=\frac{31\times {10}^{-3}}{0.4\times (20\times {10}^{-3})}=3.875\text{m/s}$

The upstream Froude number equals Fr ₁ = 8.75 (i.e. supercritical upstream flow). The downstream flow properties are deduced from the above equation:

$\frac{d_2}{d_1}=\frac{1}{2}\left(\sqrt{1+8F{r}_1^2}-1\right)=11.9$

$F{r}_2=F{r}_1\frac{2^{3/2}}{{\left(\sqrt{1+8F{r}_1^2}-1\right)}^{3/2}}=0.213$

Hence d ₂ = 0.238 m and V ₂ = 0.33 m/s.

The head loss in the hydraulic jump equals:

$\Delta H=\frac{{(d_2-d_1)}^3}{4d_1{d}_2}=0.544\text{m}$

The hydraulic power dissipated in the jump equals:

$\rho \text{gQ}\Delta \text{H}=998.2\times 9.8\times 31\times {10}^{-3}\times 0.544=165\text{W}$

where ρ is the fluid density (kg/m³), Q is in m³/s and ΔH is in m. In the laboratory flume, the dissipation power equals 165W: one 100W bulb and one 60W bulb could be lighted with the jump power.

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The Planetary Boundary Layer

James R. Holton , Gregory J. Hakim , in An Introduction to Dynamic Meteorology (Fifth Edition), 2013

8.4 Secondary Circulations and Spin Down

Both the mixed layer solution (8.17) and the Ekman spiral solution (8.26) indicate that in the planetary boundary layer the horizontal wind has a component directed toward lower pressure. As suggested by Figure 8.6, this implies mass convergence in a cyclonic circulation and mass divergence in an anticyclonic circulation, which by mass continuity requires vertical motion out of and into the boundary layer, respectively. To estimate the magnitude of this induced vertical motion, we note that if v_g = 0, the cross-isobaric mass transport per unit area at any level in the boundary layer is given by ρ ₀ v. The net mass transport for a column of unit width extending vertically through the entire layer is simply the vertical integral of ρ ₀ v. For the mixed layer, this integral is simply ρ ₀ vh(kg m⁻¹ s⁻¹), where h is the layer depth. For the Ekman spiral, it is given by

(8.30) $M=\underset{0}{\overset{De}{\int }}{\rho }_{0}vdz=\underset{0}{\overset{De}{\int }}{\rho }_0{u}_g\exp \left(-\pi z∕De\right)\sin \left(\pi z∕De\right)dz$

where $De=\pi ∕\gamma$ is the Ekman layer depth defined in Section 8.3.4.

Figure 8.6. Schematic surface wind pattern (arrows) associated with high- and low-pressure centers in the Northern Hemisphere. Isobars are shown by thin lines, and "L" and "H" designate high- and low-pressure centers, respectively.

After Stull, 1988.

Integrating the mean continuity equation (8.8) through the depth of the boundary layer gives

(8.31) $w\left(De\right)=-\underset{0}{\overset{De}{\int }}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)dz$

where we have assumed that w(0) = 0. Assuming again that v_g = 0 so that u_g is independent of x, we find after substituting from (8.26) into (8.31) and comparing with (8.30) that the mass transport at the top of the Ekman layer is given by

(8.32) ${\rho }_{0}w\left(De\right)=-\frac{\partial M}{\partial y}$

Thus, the mass flux out of the boundary layer is equal to the convergence of the cross-isobar mass transport in the layer. Noting that $-\partial u_g∕\partial y={\zeta }_g$ is just the geostrophic vorticity in this case, we find after integrating (8.30) and substituting into (8.32) that ⁴

(8.33) $w\left(De\right)={\zeta }_g\left(\frac{1}{2\gamma }\right)={\zeta }_g{\left|\frac{K_m}{2f}\right|}^{1∕2}\left(\frac{f}{\left|f\right|}\right)$

where we have neglected the variation of density with height in the boundary layer and have assumed that $1+e^{-\pi }\approx 1$ . Thus, we obtain the important result that the vertical velocity at the top of the boundary layer is proportional to the geostrophic vorticity. In this way the effect of boundary layer fluxes is communicated directly to the free atmosphere through a forced secondary circulation that usually dominates over turbulent mixing. This process is often referred to as boundary layer pumping. It only occurs in rotating fluids and is one of the fundamental distinctions between rotating and nonrotating flow. For a typical synoptic-scale system with ${\zeta }_g\sim 10^{-5}{\text{\hspace{0.17em}\hspace{0.17em}s}}^{-1}$ , $f\sim 10^{-4}{\text{\hspace{0.17em}\hspace{0.17em}s}}^{-1}$ , and $De\sim 1$ km, the vertical velocity given by (8.33) is of the order of a few millimeters per second.

An analogous boundary layer pumping is responsible for the decay of the circulation created when a cup of tea is stirred. Away from the bottom and sides of the cup there is an approximate balance between the radial pressure gradient and the centrifugal force of the spinning fluid. However, near the bottom, viscosity slows the motion and the centrifugal force is not sufficient to balance the radial pressure gradient. (Note that the radial pressure gradient is independent of depth, since water is an incompressible fluid.) Therefore, radial inflow takes place near the bottom of the cup. Because of this inflow, the tea leaves are always observed to cluster near the center at the bottom of the cup if the tea has been stirred. By continuity of mass, the radial inflow in the bottom boundary layer requires upward motion and a slow compensating outward radial flow throughout the remaining depth of the tea. This slow outward radial flow approximately conserves angular momentum, and by replacing high angular momentum fluid with low angular momentum fluid, it makes the vorticity in the cup spin down far more rapidly than could mere diffusion.

The characteristic time for the secondary circulation to spin down an atmospheric vortex is illustrated most easily in the case of a barotropic atmosphere. For synoptic-scale motions, equation (4.38) can be written approximately as

(8.34) $\frac{D{\zeta }_g}{Dt}=-f\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=f\frac{\partial w}{\partial z}$

where we have neglected ζ_g compared to f in the divergence term and have also neglected the latitudinal variation of f. Recalling that the geostrophic vorticity is independent of height in a barotropic atmosphere, (8.34) can be integrated easily from the top of the Ekman layer (z = De) to the tropopause (z = H) to give

(8.35) $\frac{D{\zeta }_g}{Dt}=+f\left[\frac{w\left(H\right)-w\left(De\right)}{\left(H-De\right)}\right]$

Substituting for w(De) from (8.33), assuming that w(H) = 0 and that $H\gg De$ , (8.35) may be written as

(8.36) $\frac{D{\zeta }_g}{Dt}=-{\left|\frac{f{K}_m}{2H^2}\right|}^{1∕2}{\zeta }_g$

This equation may be integrated in time to give

(8.37) ${\zeta }_g\left(t\right)={\zeta }_g\left(0\right)\exp \left(-t∕{\tau }_e\right)$

where ζ_g (0) is the value of the geostrophic vorticity at time t = 0, and ${\tau }_e\equiv H{\left|2∕\left(f{K}_m\right)\right|}^{1∕2}$ is the time that it takes the vorticity to decrease to e ⁻¹ of its original value.

This e-folding time scale is referred to as the barotropic spin-down time. Taking typical values of the parameters as follows: H ≡ 10 km, f = 10⁻⁴ s⁻¹, and K_m = 10 m² s⁻¹, we find that τ_e ≈ 4 days. Thus, for midlatitude synoptic-scale disturbances in a barotropic atmosphere, the characteristic spin-down time is a few days. This decay time scale should be compared to the time scale for ordinary viscous diffusion. For viscous diffusion the time scale can be estimated from scale analysis of the diffusion equation

(8.38) $\frac{\partial u}{\partial t}=K_m\frac{{\partial }^{2}u}{\partial z^2}$

If τ_d is the diffusive time scale and H is a characteristic vertical scale for diffusion, then from the diffusion equation

$U/{\tau }_d\sim K_{m}U/H^2$

so that ${\tau }_d\sim H^2∕K_m$ . For the preceding values of H and K_m , the diffusion time scale is thus about 100 days. Thus, in the absence of convective clouds the spin-down process is a far more effective mechanism than eddy diffusion for destroying vorticity in a rotating atmosphere. Cumulonimbus convection can produce rapid turbulent transports of heat and momentum through the entire depth of the troposphere. These must be considered together with boundary layer pumping for intense systems such as hurricanes.

Physically the spin-down process in the atmospheric case is similar to that described for the teacup, except that in synoptic-scale systems it is primarily the Coriolis force that balances the pressure gradient force away from the boundary, not the centrifugal force. Again, the role of the secondary circulation driven by forces resulting from boundary layer drag is to provide a slow radial flow in the interior that is superposed on the azimuthal circulation of the vortex above the boundary layer. This secondary circulation is directed outward in a cyclone so that the horizontal area enclosed by any chain of fluid particles gradually increases. Since the circulation is conserved, the azimuthal velocity at any distance from the vortex center must decrease in time or, from another point of view, the Coriolis force for the outward-flowing fluid is directed clockwise, and this force thus exerts a torque opposite to the direction of the circulation of the vortex. Figure 8.7 shows a qualitative sketch of the streamlines of this secondary flow.

Figure 8.7. Streamlines of the secondary circulation forced by frictional convergence in the planetary boundary layer for a cyclonic vortex in a barotropic atmosphere. The circulation extends throughout the full depth of the vortex.

It should now be obvious exactly what is meant by the term secondary circulation. It is simply a circulation superposed on the primary circulation (in this case the azimuthal circulation of the vortex) by the physical constraints of the system. In the case of the boundary layer, viscosity is responsible for the presence of the secondary circulation. However, other processes, such as temperature advection and diabatic heating, may also lead to secondary circulations, as shown later.

The preceding discussion has concerned only the neutrally stratified barotropic atmosphere. An analysis for the more realistic case of a stably stratified baroclinic atmosphere is more complicated. However, qualitatively the effects of stratification may be easily understood. The buoyancy force (see Section 2.7.3) will act to suppress vertical motion, since air lifted vertically in a stable environment will be denser than the environmental air. As a result, the interior secondary circulation will decrease with altitude at a rate proportional to the static stability.

This vertically varying secondary flow, shown in Figure 8.8, will rather quickly spin down the vorticity at the top of the Ekman layer without appreciably affecting the higher levels. When the geostrophic vorticity at the top of the boundary layer is reduced to zero, the pumping action of the Ekman layer is eliminated. The result is a baroclinic vortex with a vertical shear of the azimuthal velocity that is just strong enough to bring ζ_g to zero at the top of the boundary layer. This vertical shear of the geostrophic wind requires a radial temperature gradient that is in fact produced during the spin-down phase by adiabatic cooling of the air forced out of the Ekman layer. Thus, the secondary circulation in the baroclinic atmosphere serves two purposes: (1) it changes the azimuthal velocity field of the vortex through the action of the Coriolis force, and (2) it changes the temperature distribution so that a thermal wind balance is always maintained between the vertical shear of the azimuthal velocity and the radial temperature gradient.

Figure 8.8. Streamlines of the secondary circulation forced by frictional convergence in the planetary boundary layer for a cyclonic vortex in a stably stratified baroclinic atmosphere. The circulation decays with height in the interior.

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