9.4.7 Effects on kinetic energy

Although it has been built into the formalism, it is useful to explicitly show that the friction dissipates horizontal kinetic energy. For this purpose, recall that the kinetic energy for a parcel of fluid is the scalar quantity

$$2 \, K \rho \, dV \, u_{m} \, u^{m}$$ =

$$\rho \, dV \, g_{mn} \, u^{n} \, u^{m},$$ = (9.117)

where m=1,2 represents the label for the horizontal coordinates. The evolution of this energy is given by

$$\dot{K} dV \, g_{mn} \, u^{n} \, (\rho \, f^{m} + T^{mp}_{; p}),$$ = (9.118)

where mass conservation and Newton's Law were employed. Consequently, friction contributes to the evolution of kinetic energy through the term

$$dV \, g_{mn} \, u^{n} \, \tau^{mp}_{; p} \rho \, dV \, g_{mn} \, u^{n} \, F^{m}.$$ = (9.119)

The question then arises as to whether $\rho \, dV \, g_{mn} \, u^{n} \, F^{m}$ integrated over the horizontal extent of the domain is negative semi-definite, which would be the case for dissipative friction. First note that the term $u_{m} \, (\kappa \, u^{m}_{,z})_{,z}$ is not at issue here; it appears in the same form as for Cartesian coordinates and has well known dissipative properties. Some manipulations using previous results from this section yield

$$\int dV \, u_{m} \, \tau^{mp}_{; p} \int dV \, [ (u_{m} \, \tau^{mp})_{\, ; p} -\tau^{mp} \, u_{m \, ; p} ].$$ = (9.120)

The first term integrates to a boundary contribution, which vanishes with a no-slip and/or no normal stress boundary condition. Hence,

$$\int dV \, u_{m} \, \tau^{mp}_{; p} - 1/2 \, \int dV \, \tau^{mp} \, (u_{m \, ; p} + u_{p \, ; m})$$ =

$$-\int dV \, \tau^{mp} \, e_{mp},$$ = (9.121)

where the strain tensor e_mp was introduced. Use of the expression (9.101) for the transverse stress tensor leads to

$$2 \, \rho \, A \, \tau_{mp} \, e^{mp} \tau_{mp} \, (\tau^{mp} + A \, \rho \, g^{mp} \, e^{q}_{q} )$$ =

$$\tau_{mp} \, \tau^{mp},$$ = (9.122)

where $g^{mp} \, \tau_{mp} = 0$ was used. Hence, the contribution to kinetic energy from horizontal friction takes the form

$$\int dV \, u_{m} \, \tau^{mp}_{; p} - \int dV (2 \, \rho \, A)^{-1} \, \tau^{mp} \, \tau_{mp}$$ =

$$-\int \rho \, dV \, A \, (D_{T}^{2} + D_{S}^{2}),$$ = (9.123)

which shows that the kinetic energy is indeed dissipated by the chosen form of the friction, so long as the viscosity is non-negative. Since the dissipation is the scalar

$$\tau^{mn} \, \tau_{mn} = 2 (\rho \, A)^{2} \, (D_{T}^{2} + D_{S}^{2}),$$ (9.124)

it is coordinate invariant. Again, note that the indices m,n extend only over the horizontal directions m,n = 1,2. It is convenient here to point out a connection between the rate of kinetic energy dissipation and the Smagorinsky formulation for viscosity. In the Smagorinsky (1963) scheme (Section 33.7), viscosity A is determined as a function of the total amount of horizontal strain in the flow, as well as the grid spacing. By ``total amount of strain'', Smagorinsky means the scalar quantity

$$2 \, (2 \, \rho \, A)^{-2} \, \tau^{mn} \, \tau_{mn}$$ D² =

= D_T² + D_S². (9.125)

That is, |D| represents the total rate of horizontal strain for the resolved motions. As D is constructed as a scalar quantity, its value is the same in any set of horizontal curvilinear coordinates. Hence, from a mathematical and numerical perspective, D is a sensible quantity to use for constructing viscosity.