Manifold documentation mostly finished

Author: baperry2

Docs/sphinx/Spray.rst

@@ -2,9 +2,9 @@
 
 .. _Spray:
 
-*****
-Spray
-*****
+=======
+ Spray
+=======
 
 Spray Equations
 ===============
@@ -163,7 +163,7 @@ The procedure is as follows for updating the spray droplet:
    .. math::
       Y_{r,i} = \left\{\begin{array}{c l}
       \displaystyle Y_{v,i} + A (Y_{g,i} - Y_{v,i}) & {\text{if $i \in \mathcal{S}_{pc}$ and $Y_{v,i} > 0$}}, \\
-      \displaystyle\frac{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{v,k}}{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{g,k}} Y_{g,i} & {\text{otherwise}},
+      \displaystyle\frac{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{r,k}}{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{g,k}} Y_{g,i} & {\text{otherwise}},
       \end{array}\right. \quad \forall\; i \in \mathcal{S}_g.
 
 #. The average molar mass, specific heat, and density of the reference state in the gas film are computed as
@@ -347,7 +347,7 @@ The modified procedure for Manifold-based gas phase chemistry is as follows for
    .. math::
       Y_{r,i} = \left\{\begin{array}{c l}
       \displaystyle Y_{v,i} + A (Y_{g,i} - Y_{v,i}) & {\text{if $i \in \mathcal{S}_{pc}$ and $Y_{v,i} > 0$}}, \\
-      \displaystyle\frac{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{v,k}}{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{g,k}} Y_{g,i} & {\text{otherwise}},
+      \displaystyle\frac{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{r,k}}{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{g,k}} Y_{g,i} & {\text{otherwise}},
       \end{array}\right. \quad \forall\; i \in \mathcal{S}_g.
 
    and note that :math:`\xi_i = W_{ij} Y_j`. We note that :math:`Y_{g,i} \: \forall i \in \mathcal{S}_g` can be evaluated from the manifold, but to reduce the number of variables that must be
@@ -359,11 +359,47 @@ The modified procedure for Manifold-based gas phase chemistry is as follows for
       \displaystyle 0.0 & {\text{otherwise}},
       \end{array}\right. \quad \forall\; i \in \mathcal{S}_g.
 
-   and its complement :math:`Y^{nc}_{g,i}` such that :math:`Y^{pc}_{g,i} + Y^{nc}_{g,i} = Y_{g,i}`, as well as similar definitions for :math:`Y^{nc}_{r,i}`, :math:`Y^{pc}_{r,i}`,
-   :math:`Y^{nc}_{v,i}`, and  :math:`Y^{pc}_{v,i}`.
+   and its complement :math:`Y^{nc}_{g,i}` such that :math:`Y^{pc}_{g,i} + Y^{nc}_{g,i} = Y_{g,i}`, as well as similar definitions for :math:`Y^{nc}_{r,i}, Y^{pc}_{r,i},
+   Y^{nc}_{v,i}, \text{and } Y^{pc}_{v,i}`. With these definitions,
+
+   .. math::
+      Y^{pc}_{r,i} &=  Y^{pc}_{v,i} + A (Y^{pc}_{g,i} - Y^{pc}_{v,i}), \\
+      Y^{nc}_{r,i} &= \theta Y^{nc}_{g,i}, \\
+      \text{where } \theta &= \frac{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{r,k}}{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y_{g,k}}
+                            = \frac{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y^{pc}_{r,k}}{1 - \sum_{k \in \mathcal{S}_{pc} | Y_{v,k} > 0 } Y^{pc}_{g,k}}
+
+
+   Therefore, in order to compute :math:`\xi_{r,j} = W_{ij} Y_{r,i}` from the available data :math:`(\xi_{g,j}, Y^{pc}_{g,i}(\xi_{g,j}), \text{and } Y^{pc}_{v,i} )`, we note
+
+   .. math::
+      \xi_{r,j} &= W_{ij} (Y^{pc}_{r,i} + Y^{nc}_{r,i}) = W_{ij} Y^{pc}_{r,i} + W_{ij} \theta Y^{nc}_{g,i}, \text{and} \\
+      \xi_{g,j} &= W_{ij} (Y^{pc}_{g,i} + Y^{nc}_{g,i}) \therefore W_{ij} Y^{nc}_{g,i} = \xi_{g,j} - W_{ij} Y^{pc}_{g,i}.
+
+   We can therefore compute :math:`\xi_{r,j}` from only available data using:
+
+   .. math::
+      \xi_{r,j} =  W_{ij} Y^{pc}_{r,i} + \theta (\xi_{g,j} - W_{ij} Y^{pc}_{g,i} ).
+
+   The present implementation specializes to the case (common in combustion modeling) where the manifold parameters :math:`\xi_j` are either progress variable like :math:`(W_{ij}=0 \; \forall\; i \in \mathcal{S}_{pc})`
+   or mixture fraction like (:math:`W_{ij}=1` for one :math:`i \in \mathcal{S}_{pc}` and :math:`0` for all others).
+
+#. Proceeds as in the ideal gas implementation, but :math:`\bar{M}_r` and :math:`\rho_r` are computed as functions of the manifold :math:`\phi(\xi_j))`.
+   :math:`c_{p,r}` is not computed because the energy equation is not solved.
+
+#. Again, :math:`\mu_r`, :math:`\lambda_r`, and :math:`\rho D_{r,n}`  are computed as functions of the manifold :math:`\phi(\xi_j))`.
+
+#. Diffusion coefficent modification proceeds as in the detailed chemistry case.
+
+#. Momentum source term computation proceeds as in the detailed chemistry case.
+
+#. Mass source term proceeds as in the detailed chemistry case. Energy source term is ignored.
+
+#. Gas phase source terms follow the detailed chemistry implementation, except that the energy/enthalpy equations are ignored and
+
+   .. math::
+      S_{\rho \xi_j} = W_{ij} S_{\rho Y_i} = W_{ij} \mathcal{C} \sum_{n \in \mathcal{S}_L} \mathbf{L}_{i,n}\dot{m}_n \quad \forall\; i \in \mathcal{S}_{g},
+
 
-   Therefore, in order to compute :math:`\xi_{r,j} = W_{ij} Y_{r,j}` from the available data (:math:`\xi_{g,i}`,  :math:`Y^{pc}_{g,i}(\xi_{g,i})`, and :math:`Y^{pc}_{v,i}` ), we note
-   :math:`\xi_{r,j} = W_{ij} (Y^{nc}_{r,j} + Y^{pc}_{r,j})`
 
 Spray Flags and Inputs
 ======================
@@ -445,7 +481,7 @@ and the *FuelLib-based GCM* are outlined in the subsections below. Please note t
 The source code for the liquid spray properties can be found in ``SprayProperties.H``.
 
 PeleMP Implementation
-^^^^^^^^^^^^^^^^^^^^^
+~~~~~~~~~~~~~~~~~~~~~
 
 .. table::
 
@@ -510,7 +546,7 @@ PeleMP Implementation
 
 
 FuelLib-Based GCM
-^^^^^^^^^^^^^^^^^
+~~~~~~~~~~~~~~~~~
 
 Currently the *GCM* approach of estimating liquid fuel properties is only available in PeleLMeX and requires: