Search:
D68 /

# EquationApproachSimilitude

The Equation Approach presumes that there is a valid mathemetical model interconnecting all relevant phenomena under consideration. For non-reactive compressible multi-component flows this would be the set of Navier-Stokes-Equations including the mass, momentum, energy and species conservation.

$${ { \partial \rho } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho \vec{ v } ) = 0$$ $${ { \partial \rho \vec{ v } } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho \vec{ v } \otimes \vec{ v } + p I ) = \rho \vec{ g } + \vec{ \nabla } \cdot \tau$$ $${ { \partial \rho e_t } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho \vec{ v } h_t ) = \vec{ \nabla } \cdot ( - \vec{ q } ) + \vec{ \nabla } \cdot ( \tau \vec{ v } ) + \rho \vec{ g } \cdot \vec{ v }$$ $${ { \partial \rho Y_i } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho ( \vec{ v } + {\vec{ v }}_{d,i} ) Y_i ) = 0$$

This is a set of partial differential equations not easily solved for real problems.

Even in studying the character of these equations it helps to reformulate these equations in a dimensionless form. Additional motivation for doing this is the knowledge that the set of used units shall not affect the validity of these basic conservation laws and that, secondly, nature knows only little about human created units like "meter", "second" and "kilogram".

Instead we choose natural scales for the length, time, etc, provided by the problem itself.

$$l = L l^*; t = T t^*; \rho = \rho_r \rho^* p = P p^*$$

As a derived quantity the velocities should be scaled by: $$v = V v^* = {L \over T} v^*$$

The scales or representative quantities are indicated by capital letters or by the subscript r. Inserting this in the Navier-Stokes equations gives

$${ { \partial \rho^* } \over { \partial t^* } } + \vec{ \nabla^* } \cdot ( \rho^* \vec{ v^* } ) = 0$$ $${ { \partial \rho^* \vec{ v^* } } \over { \partial t^* } } + \vec{ \nabla^* } \cdot ( \rho^* \vec{ v^* } \otimes \vec{ v^* } + {P \over \rho_r V^2 } p^* I ) = {L g \over V^2 } \rho^* {\vec{ e }}_g + {\mu_r \over \rho_r L V } \Delta^* v^*$$ $${c_{v,r} \over c_{p,r} } { { \partial \rho^* e_t^* } \over { \partial t^* } } + \vec{ \nabla^* } \cdot ( \rho^* \vec{ v^* } h_t^* ) = - { \lambda_r \over \rho_r c_{p,r} L V } \Delta T + { \mu_r V^2 \over V L \rho_r c_{p,r} T_r } \Delta + \rho \vec{ g } \cdot \vec{ v }$$ $${ { \partial \rho Y_i } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho ( \vec{ v } + {\vec{ v }}_{d,i} ) Y_i ) = 0$$