Formulae to describe porous flow

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FORMULAE TO DESCRIBE POROUS FLOW

Chapter 3

3 Summary and conclusions.

For the description of porous flow the Forchheimer equation is normally used. Several formulae have been proposed for the coefficientsCland c2from this equation. All these formulae are based on experiments. Those coefficients represent the friction and resistance caused by the porous medium. The Forchheimer equation is a somewhat semi-empirical formula. The phenomena that appear in a porous flow are not really described and implemented in the formula.

Inthis report the assumptions and approximations to derive the Forchheimer equation using the Navier-Stokes equation were mentioned. To describe the flow on a porous structure (e.g., a breakwater) and the internal flow with a numerical model, the Navier-Stokes equation need to be adapted for the internal flow. In this report the Navier-Stokes equation is adapted for porous flow. The approach to derive this equation gives the adapted Navier-Stokes equation anditshows how several parameters are present in the coefficients. Those coefficients are also present in the Forchheimer equation. So, it gives expressions for the coefficients for the Forchheimer equation as well. The derived formula can be used for a numerical model that describes the flow on a porous structure (with Navier-Stokes equations) and the flow in the structure. This formula is derived by regarding several contributions of the force on the grains. Each contribution is written as a result of one term from the Navier-Stokes equation. Sometimes rather rough assumptions and approximations were made to derive the adapted Navier-Stokes equation for porous flow; those were mentioned in earlier sections.

The contributions to the total force on the grains that are included in this formula are called laminar friction, turbulence friction, added mass and

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3. SUMMARY AND CONCLUSIONS

resistance as a result of the convective term (shape-resistance). The assumption is made that the force on the grains can be described by a summation of these contributions. Turbulence friction is not represented, or is not clearly represented, in existing derivations of formulae for porous flow. This friction is normally included in one term together with resistance as a result of the convective term (shape resistance). Those two contributions together, are usually called "drag".

The problem of finding suitable values for the coefficients that describe the several processes, has to be solved by doing measurements. Most of the measurements were done under stationary flow conditions. Measurements for determining values of coefficients for non-stationary flow conditions are not sufficient yet to find suitable values for CM' The other coefficients are

probably different under non-stationary flow conditions. This may cause that measurements of coefficients for Cl and _C2under stationary flow conditions can not be used for Cl and _C2 under non-stationary flow conditions. The coefficients are probably depending on the Re-number and the KC-number. The formula for the one-dimensional porous flow is given in equation 52. The equations that can describe two-dimensional porous flow can be written as:

( l+c_M ) aUj ( l+c

v)

aUj2 ( 1+c

v)

_aUjWj₌

+ + ng at n2 g ax n2 g az ₍₅₃₎ 1 ap 1 l5 Q x - c - - u - c_T- - 2 Uj

I

Ujrep

I

+ Pw g aX L _g _II2 j _g _l4 g

I

( 1_{+CM )} aWj ng at 1 ap P_w g az (54) 47

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