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# 物理代写|空气动力学代写Aerodynamics代考|EAS4101 Symmetric Form

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## 物理代写|空气动力学代写Aerodynamics代考|Symmetric Form

Consider the equations of one-dimensional flow in primitive variables:
$$\begin{array}{r} \frac{\partial \rho}{\partial t}+u \frac{\partial \rho}{\partial x}+\rho \frac{\partial u}{\partial x}=0 \ \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+\frac{1}{\rho} \frac{\partial p}{\partial x}=0 \ \frac{\partial p}{\partial t}+\rho c^{2} \frac{\partial u}{\partial x}+u \frac{\partial p}{\partial x}=0 \end{array}$$

Then subtracting the first equation multiplied by $c^{2}$ from the third equation, we find that
$$\frac{\partial p}{\partial t}-c^{2} \frac{\partial \rho}{\partial t}+u\left(\frac{\partial p}{\partial x}-c^{2} \frac{\partial \rho}{\partial x}\right)=0$$
This is equivalent to a statement that the entropy
$$S=\log \left(\frac{p}{\rho^{\gamma}}\right)=\log p-\gamma \log \rho$$
is constant since
$$d S=\frac{d p}{p}-\gamma \frac{d \rho}{\rho}=\frac{1}{p}\left(d p-c^{2} d \rho\right)$$

## 物理代写|空气动力学代写Aerodynamics代考|Riemann Invariants

In the case of one-dimensional isentropic flow, these equations reduce to
$$\frac{2}{\gamma-1} \frac{\partial c}{\partial t}+\frac{2 u}{\gamma-1} \frac{\partial c}{\partial x}+c \frac{\partial u}{\partial x}=0$$

\begin{aligned} \frac{\partial u}{\partial t}+\frac{2 c}{\gamma-1} \frac{\partial c}{\partial x}+u \frac{\partial u}{\partial x} &=0 \ \frac{\partial S}{\partial t}+u \frac{\partial S}{\partial x} &=0 \end{aligned}
Now the first two equations can be added and subtracted to yield
$$\frac{\partial R^{+}}{\partial t}+(u+c) \frac{\partial R^{+}}{\partial x}=0$$
and
$$\frac{\partial R^{-}}{\partial t}+(u-c) \frac{\partial R^{-}}{\partial x}=0,$$
where $R^{+}$and $R^{-}$are the Riemann invariants
$$R^{+}=u+\frac{2 c}{\gamma-1}, R^{-}=u-\frac{2 c}{\gamma-1},$$
which remain constant as they are transported at the wave speeds $u+c$ and $u-c$. The Riemann invariants prove to be useful in the formulation of far field boundary conditions designed to minimize wave reflection.

## 物理代写|空气动力学代写Aerodynamics代考|Symmetric Form

$$\frac{\partial \rho}{\partial t}+u \frac{\partial \rho}{\partial x}+\rho \frac{\partial u}{\partial x}=0 \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+\frac{1}{\rho} \frac{\partial p}{\partial x}=0 \frac{\partial p}{\partial t}+\rho c^{2} \frac{\partial u}{\partial x}+u \frac{\partial p}{\partial x}=0$$

$$\frac{\partial p}{\partial t}-c^{2} \frac{\partial \rho}{\partial t}+u\left(\frac{\partial p}{\partial x}-c^{2} \frac{\partial \rho}{\partial x}\right)=0$$

$$S=\log \left(\frac{p}{\rho^{\gamma}}\right)=\log p-\gamma \log \rho$$

$$d S=\frac{d p}{p}-\gamma \frac{d \rho}{\rho}=\frac{1}{p}\left(d p-c^{2} d \rho\right)$$

## 物理代与写|空气动力学代写Aerodynamics代考|Riemann Invariants

$$\begin{gathered} \frac{2}{\gamma-1} \frac{\partial c}{\partial t}+\frac{2 u}{\gamma-1} \frac{\partial c}{\partial x}+c \frac{\partial u}{\partial x}=0 \ \frac{\partial u}{\partial t}+\frac{2 c}{\gamma-1} \frac{\partial c}{\partial x}+u \frac{\partial u}{\partial x}=0 \frac{\partial S}{\partial t}+u \frac{\partial S}{\partial x} \quad=0 \end{gathered}$$

$$\frac{\partial R^{+}}{\partial t}+(u+c) \frac{\partial R^{+}}{\partial x}=0$$

$$\frac{\partial R^{-}}{\partial t}+(u-c) \frac{\partial R^{-}}{\partial x}=0,$$

$$R^{+}=u+\frac{2 c}{\gamma-1}, R^{-}=u-\frac{2 c}{\gamma-1},$$

## MATLAB代写

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