A consideration on the effects of drag force acting on roughness element - The relationship between drag coefficient and dissipation rate

Momentum and energy budgets on an urban surface are investigated based on theoretical approach by deriving the budget equations. Firstly, we have introduced the integral equation for energy, which corresponds to the equation as known as Karman's integral law for momentum. These equations can explain the effect of drag force acting on roughness elements on the budgets of momentum and energy. Secondly, it is shown that the additional term appears to explain for the drag force to be able to directly absorb momentum provided out of boundary layer. Although the corresponding term does not exist in the energy budget equations, the drag force may work to increase the turbulence energy through the additional turbulence production terms. Thirdly, the relationship between drag coefficient and dissipation rate have been discussed. These quantities have positive correlation and can be an index to explain the wind velocity deficit.