Reservoir Farend Truncated Boundary Condition

The truncated boundary for the finite element and boundary element modeling of the infinite reservoir was worked by so many researchers. Sommerfeld boundary condition is the most common one that is based on the assumption that at long distance from the dam face, water wave can be considered as plane wave. A plane wave can be represented by an equation of the form:

in which n is the normal at the truncation boundary. This represents well-known Sommerfeld radiation condition. It introduces a damping in the system and models the loss of energy in the outgoing waves. Under this condition p = F (x - Ct)

This equation leads to the following condition:

for the far end reservoir, the classical solution for the hydrodynamic pressure was obtained in which there are in-phase and out-of-phase pressures. As a rule for the truncated boundary there is no reflection for the outgoing wave and all of our effort is in modeling the energy loss in outgoing wave such that all energy can be absorbed on the truncated boundaries. The solution to the wave equation can be investigated for a rigid dam with rectangular reservoir subjected to horizontal ground motion and the following boundary conditions:

ay dy dX = —p u at x = 0 dp = dp = _ 1 dp at x = L

dx on C dt n/"» r\T,£»conT'£i iV\T* o Viormrvnip rrr/MTn/'l ^vv^i + o r\f u

The hydrodynamic pressure for a harmonic ground excitation of ug = elwt would be:

4p n to

It was found that the Sommerfeld radiation damping condition does not yield good approximation for the infinite reservoir for excitation frequencies between the first and second natural frequencies of the reservoir. Within this range of frequencies an increase in the length of reservoir does not lead to any increase in the accuracy(Humar and Roufaiel, 1983).

Humar and Roufaiel (1983) used a radiation condition which adequately models the loss over a wide range of excitation frequencies, also it was shown that the conditions give much better results as compared to the plane wave Sommerfeld condition. Their radiation condition derived only for the horizontal ground motion in the case of dam with vertical up-stream face. The Solution of the equation 2.41 when ug = elwt can be represented as following :

m=mi to

m=mi

A2 U2

Jut n

It was already shown that at large distance from the dam, the second term in the equation vanishes, and the pressure is given as:

m1-1 -( /A2 _ ^ x m1-1 p(x,y,t)=Y, AmeV> m ^ cos \my e™ = £ pm (2.46)

From the above equation we can write:

For the case of m = 2 (when u lies between ^1and u2) equation 2.46 gives p = p1 and equation 2.47 becomes:

Therefore, it can be said that the following condition should be applied at the truncated boundary:

Apparently, the preceding condition is not exact when u is greater than u2. For large value of u equation 2.49 reduces to equation 2.43.

The hydrodynamic pressure obtained by applying the modified boundary condition at x = L represented by equations 2.48 and 2.49 will be given by an expression similar to equation 2.44, except that Zmwill now be as follows:

The modified boundary condition at the far end was obtained with ignoring the second term in the pressure equation at the large distance from the dam. It was found that the modified radiation boundary condition gives much better results as compared to the Sommerfeld one for excitation frequencies between 0.0 and 2.

Sharan (1984) showed that the condition of |n = 0 for incompressible fluid at the far end is the form of that for a rigid stationary boundary and the behavior of the fluid motion at the truncated boundary is not truly presented. A large extent of the fluid domain is required to be included in the analysis. The above condition is same as Sommerfeld condition for the incompressible fluid (C = œ>). Under the assumption of incompressible fluid in the rectangular reservoir and rigid dam, the governing equation of the pressure would be the Laplace equation. Knowing the pressure equation, he found that at large distance from upstream face condition would be:

It can be observed that at very large distance away from the dam face p = 0.0 therefore ^ =0 which is same condition at the infinity for the infinite an J

reservoir. He found that under the assumption of the rigid dam and rigid rectangular reservoir bottom, for the horizontal vibration. the truncated boundary can be located very close to the structure. It was found that although the proposed boundary condition was derived for the dam with the vertical upstream face, the results for an inclined face are relatively more accurate than those for a vertical face.

Sharan (1985) used the radiation condition for the submerged structure surrounded by unbounded extent of compressible fluid as following:

Deferent geometry of the solid-fluid interface was used to find the accuracy of the proposed radiation boundary condition.

Sharan (1987) proposed a damper radiation boundary condition for the time domain analysis of the compressible fluid with small amplitude. The structure submerged in unbounded fluid in the upstream direction. The proposed radiation condition was:

The above boundary condition was found to be very effective and efficient for a wide range of the excitation frequency. For the finite value of c and h, the effectiveness of the proposed damper depends on the period of excitation T, ( 2t ). If the value of T be near the natural period of vibration of the fluid domain (TC ^ 4), the magnitude of pand dp become infinitely large. In this case approximated boundary condition for the truncated surface may be written as „p = 0 . Therefore for such a case, neither Sommerfeld damper nor the proposed one would be effective. In the case of TC ¿ 4, the magnitude of , pCC, becomes small compared to that of |p. For the limiting case p ^ 0 when T ^ 0. On the other hand, for TC ^ 4, dp is much less than that of pC. For the limiting case dp ^ 0 when T ^to. Thus in both case, one of the terms on the right hand side of proposed equation of damping would be small compare to the other. As a general, it was found that proposed boundary condition at the truncated surface is more efficient and needs less computational time. In the case of TC = 1 (high excitation frequency) both methods shows discrepancy with the case of L = to. It is not an important problem because in high frequency excitation the hydrodynamic forces are not comparable so that its error in hydrodynamic pressure is not significant.

Sharan (1985) expressed the condition at the truncated face as:

For (fi = C) less than which is for excitation frequency less than the second natural frequency of the reservoir if p is sufficiently large, then a = 0 and \$ = -p(2h)2 - 1 and for | < fi < , @ = 0 and a = p1 - (^)2. For fi < n, In is function of p instead of dp, therefore Sommerfeld radiation boundary condition is not justified for fi < |. For these range of frequencies values of p,dp ,and ^p approach zeroes as the ratio | is increases. This explains why satisfactory results may be obtained even with use of improper boundary condition such as d^p =0 and dp = — C |p by considering a very large extent of the reservoir. For fi > even for large value of | , the parameters a and \$ are sensitive to x and y coordinate point on the truncated surface.

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