| 950 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 951 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
| 952 |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 953 |
= \frac{ \vec{\bf v}^* }{ \Delta t } |
= \frac{ \vec{\bf v}^{n} }{ \Delta t } |
| 954 |
|
+ \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 955 |
|
+ {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
| 956 |
\end{eqnarray*} |
\end{eqnarray*} |
| 957 |
\begin{eqnarray} |
\begin{eqnarray} |
| 958 |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
| 961 |
= \epsilon_{fw} (P-E) |
= \epsilon_{fw} (P-E) |
| 962 |
\label{eq:eta-n+1-CrankNick} |
\label{eq:eta-n+1-CrankNick} |
| 963 |
\end{eqnarray} |
\end{eqnarray} |
| 964 |
where: |
We set |
| 965 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 966 |
\vec{\bf v}^* & = & |
\vec{\bf v}^* & = & |
| 967 |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
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