Benutzer:Aerospace engineer/Formelsammlung Strömungslehre

Grundlagen

Dehngeschwindigkeiten

$\epsilon _{ii}={\frac {\partial v_{i}}{\partial x_{i}}}=D_{ii}\quad i=1,2,3$

${\underline {\epsilon }}={\begin{pmatrix}{\frac {\partial u}{\partial x}}\\{\frac {\partial v}{\partial y}}\\{\frac {\partial w}{\partial z}}\end{pmatrix}}$

$\epsilon _{x}=\epsilon _{11}$

$\epsilon _{y}=\epsilon _{22}$

$\epsilon _{z}=\epsilon _{33}$

Schergeschwindigkeiten/Winkelgeschwindigkeiten der Scherung

$\vartheta _{ij}=\vartheta _{ji}={\frac {1}{2}}\left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)=D_{ij}=D_{ji}\quad i,j=1,2,3\quad i\neq j$

${\underline {\vartheta }}={\frac {1}{2}}{\begin{pmatrix}{\frac {\partial v}{\partial z}}+{\frac {\partial w}{\partial y}}\\{\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}}\\{\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial y}}\end{pmatrix}}$

$\vartheta _{x}=\vartheta _{23}=\vartheta _{32}$

$\vartheta _{y}=\vartheta _{13}=\vartheta _{31}$

$\vartheta _{z}=\vartheta _{12}=\vartheta _{21}$

Winkelgeschwindigkeit/Wirbelvektor

${\underline {\omega }}={\frac {1}{2}}\operatorname {rot} \ {\underline {v}}={\frac {1}{2}}{\begin{pmatrix}{\frac {\partial w}{\partial y}}-{\frac {\partial v}{\partial z}}\\{\frac {\partial u}{\partial z}}-{\frac {\partial w}{\partial x}}\\{\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}\end{pmatrix}}$

Tensor der Rotation

$\omega _{ij}=-\omega _{ji}={\frac {1}{2}}\left({\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {\partial v_{i}}{\partial x_{j}}}\right)=-W_{ij}=W_{ji}\quad i,j=1,2,3$

$\omega _{x}=\omega _{23}$

$\omega _{y}=\omega _{31}$

$\omega _{z}=\omega _{12}$

Volumenausdehnung/Volumendilatation

${\frac {d(\Delta V)}{\Delta V}}$ oder ${\frac {D(\Delta V)}{\Delta V}}$

Kontinuitätsgleichungen

Kontinuitätsgleichung in konservativer integraler Form

${\frac {\partial }{\partial t}}\iiint \limits _{V}\rho dV+\iint \limits _{S}\rho {\underline {v}}\cdot {\underline {dS}}=0$

Impulsgleichung

Deformationsgradient

$F_{ik}={\frac {\partial x_{i}}{\partial \xi _{k}}}\quad i,k=1,2,3$

${\underline {\underline {F}}}={\begin{pmatrix}{\frac {\partial x_{1}}{\partial \xi _{1}}}&{\frac {\partial x_{1}}{\partial \xi _{2}}}&{\frac {\partial x_{1}}{\partial \xi _{3}}}\\{\frac {\partial x_{2}}{\partial \xi _{1}}}&{\frac {\partial x_{2}}{\partial \xi _{2}}}&{\frac {\partial x_{2}}{\partial \xi _{3}}}\\{\frac {\partial x_{3}}{\partial \xi _{1}}}&{\frac {\partial x_{3}}{\partial \xi _{2}}}&{\frac {\partial x_{3}}{\partial \xi _{3}}}\end{pmatrix}}$

Spannungstensor

$\sigma _{xx}=-p+\tau _{xx}$

$\sigma _{yy}=-p+\tau _{yy}$

$\sigma _{zz}=-p+\tau _{zz}$

${\underline {\underline {\sigma }}}={\begin{pmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\end{pmatrix}}$

Impulsgleichung in nicht-konservativer differentieller Form

in Komponentenschreibweise

x-Komponente: $\rho {\frac {Du}{Dt}}=-{\frac {\partial p}{\partial x}}+{\frac {\partial \tau _{xx}}{\partial x}}+{\frac {\partial \tau _{yx}}{\partial y}}+{\frac {\partial \tau _{zx}}{\partial z}}+\rho f_{x}$

y-Komponente: $\rho {\frac {Dv}{Dt}}=-{\frac {\partial p}{\partial y}}+{\frac {\partial \tau _{xy}}{\partial x}}+{\frac {\partial \tau _{yy}}{\partial y}}+{\frac {\partial \tau _{zy}}{\partial z}}+\rho f_{y}$

z-Komponente: $\rho {\frac {Dw}{Dt}}=-{\frac {\partial p}{\partial z}}+{\frac {\partial \tau _{xz}}{\partial x}}+{\frac {\partial \tau _{yz}}{\partial y}}+{\frac {\partial \tau _{zz}}{\partial z}}+\rho f_{z}$

in Vektorschreibweise

$\rho {\frac {D{\underline {v}}}{Dt}}=\operatorname {div} {\underline {\underline {\sigma }}}+\rho {\underline {f}}$

Impulsgleichung in konservativer differentieller Form

in Komponentenschreibweise

x-Komponente: ${\frac {\partial (\rho u)}{\partial t}}+\nabla \cdot (\rho u{\underline {v}})=-{\frac {\partial p}{\partial x}}+{\frac {\partial \tau _{xx}}{\partial x}}+{\frac {\partial \tau _{yx}}{\partial y}}+{\frac {\partial \tau _{zx}}{\partial z}}+\rho f_{x}$

y-Komponente: ${\frac {\partial (\rho v)}{\partial t}}+\nabla \cdot (\rho v{\underline {v}})=-{\frac {\partial p}{\partial y}}+{\frac {\partial \tau _{xy}}{\partial x}}+{\frac {\partial \tau _{yy}}{\partial y}}+{\frac {\partial \tau _{zy}}{\partial z}}+\rho f_{y}$

z-Komponente: ${\frac {\partial (\rho w)}{\partial t}}+\nabla \cdot (\rho w{\underline {v}})=-{\frac {\partial p}{\partial z}}+{\frac {\partial \tau _{xz}}{\partial x}}+{\frac {\partial \tau _{yz}}{\partial y}}+{\frac {\partial \tau _{zz}}{\partial z}}+\rho f_{z}$

in Vektorschreibweise

${\frac {\partial (\rho {\underline {v}})}{\partial t}}+\operatorname {div} (\rho \cdot {\underline {v}}\otimes {\underline {v}})=\operatorname {div} {\underline {\underline {\sigma }}}+\rho {\underline {f}}$

Eulergleichungen

Spannungstensor

${\underline {\underline {\sigma }}}={\begin{pmatrix}-p&0&0\\0&-p&0\\0&0&-p\end{pmatrix}}$

Eulergleichungen in nicht-konservativer differentieller Form

in Komponentenschreibweise

x-Komponente: $\rho {\frac {Du}{Dt}}=-{\frac {\partial p}{\partial x}}+\rho f_{x}$

y-Komponente: $\rho {\frac {Dv}{Dt}}=-{\frac {\partial p}{\partial y}}+\rho f_{y}$

z-Komponente: $\rho {\frac {Dw}{Dt}}=-{\frac {\partial p}{\partial z}}+\rho f_{z}$

in Vektorschreibweise

$\rho {\frac {D{\underline {v}}}{Dt}}=-\operatorname {grad} \ p+\rho {\underline {f}}$

Eulergleichungen in konservativer differentieller Form

in Komponentenschreibweise

x-Komponente: ${\frac {\partial (\rho u)}{\partial t}}+\nabla \cdot (\rho u{\underline {v}})=-{\frac {\partial p}{\partial x}}+\rho f_{x}$

y-Komponente: ${\frac {\partial (\rho v)}{\partial t}}+\nabla \cdot (\rho v{\underline {v}})=-{\frac {\partial p}{\partial y}}+\rho f_{y}$

z-Komponente: ${\frac {\partial (\rho w)}{\partial t}}+\nabla \cdot (\rho w{\underline {v}})=-{\frac {\partial p}{\partial z}}+\rho f_{z}$

in Vektorschreibweise

${\frac {\partial (\rho {\underline {v}})}{\partial t}}+\operatorname {div} (\rho \cdot {\underline {v}}\otimes {\underline {v}})=-\operatorname {grad} \ p+\rho {\underline {f}}$

Navier-Stokes Gleichungen

Symmetrischer Tensor der Deformationsgeschwindigkeit/Verzerrungsgeschwindigkeit

$D_{ij}=D_{ji}={\frac {1}{2}}\left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)=\vartheta _{ij}=\vartheta _{ji}\quad D_{ii}={\frac {\partial v_{i}}{\partial x_{i}}}=\epsilon _{ii}\quad i=1,2,3$

${\underline {\underline {D}}}={\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {1}{2}}\left({\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}\right)&{\frac {1}{2}}\left({\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}}\right)\\{\frac {1}{2}}\left({\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}\right)&{\frac {\partial v}{\partial y}}&{\frac {1}{2}}\left({\frac {\partial v}{\partial z}}+{\frac {\partial w}{\partial y}}\right)\\{\frac {1}{2}}\left({\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}}\right)&{\frac {1}{2}}\left({\frac {\partial v}{\partial z}}+{\frac {\partial w}{\partial y}}\right)&{\frac {\partial w}{\partial z}}\end{pmatrix}}={\begin{pmatrix}\epsilon _{11}&\vartheta _{12}&\vartheta _{13}\\\vartheta _{21}&\epsilon _{22}&\vartheta _{23}\\\vartheta _{31}&\vartheta _{32}&\epsilon _{33}\end{pmatrix}}$

Antisymmetrischer Tensor der Drehgeschwindigkeit

$W_{ij}=-W_{ji}={\frac {1}{2}}\left({\frac {\partial v_{i}}{\partial x_{j}}}-{\frac {\partial v_{j}}{\partial x_{i}}}\right)=-\omega _{ij}=\omega _{ji}\quad W_{ii}=0\quad i=1,2,3$

${\underline {\underline {W}}}={\begin{pmatrix}0&{\frac {1}{2}}\left({\frac {\partial u}{\partial y}}-{\frac {\partial v}{\partial x}}\right)&{\frac {1}{2}}\left({\frac {\partial u}{\partial z}}-{\frac {\partial w}{\partial x}}\right)\\{\frac {1}{2}}\left({\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}\right)&0&{\frac {1}{2}}\left({\frac {\partial v}{\partial z}}-{\frac {\partial w}{\partial y}}\right)\\{\frac {1}{2}}\left({\frac {\partial w}{\partial x}}-{\frac {\partial u}{\partial z}}\right)&{\frac {1}{2}}\left({\frac {\partial w}{\partial y}}-{\frac {\partial v}{\partial z}}\right)&0\end{pmatrix}}$

Tensor der Deformationsgeschwindigkeit/Verzerrungsgeschwindigkeit

${\dot {F}}_{ik}={\frac {\partial ^{2}x_{i}({\underline {\xi }},t)}{\partial \xi _{k}\partial t}}={\frac {\partial v_{i}}{\partial \xi _{k}}}=\sum _{l=1}^{3}{\frac {\partial v_{i}}{\partial x_{l}}}{\frac {\partial x_{l}}{\partial \xi _{k}}}\quad i,k=1,2,3$

${\underline {\underline {\dot {F}}}}=(\operatorname {grad} \ {\underline {v}})^{T}{\underline {\underline {F}}}$

$\left.{\underline {\underline {\dot {F}}}}\right|_{t_{b}=t}=\left.{\underline {\underline {\dot {F}}}}\right|_{{\underline {\underline {F}}}={\underline {\underline {I}}}}=(\operatorname {grad} \ {\underline {v}})^{T}={\underline {\underline {D}}}+{\underline {\underline {W}}}$