Formelsammlung Gasdynamik

0. Einleitung

Kompressible Strömung: $\rho = \rho \left( p \right)$
Dichteveränderliche Strömung: $\rho = \rho \left( {p,T} \right)$
Maß für die Kompressibilität: $\left| {\frac{{\Delta \rho }}{\rho }} \right| = \left| {\frac{{M{a^2}}}{2}} \right|$
Machzahl: $Ma = \frac{u}{c}$ ; $c = \sqrt {\kappa RT}$
Gradient: $\operatorname{grad} p = \nabla p = \left( {\begin{array}{*{20}{c}}{\frac{{\partial p}}{{\partial x}}} \\ {\frac{{\partial p}}{{\partial y}}} \\ {\frac{{\partial p}}{{\partial z}}} \end{array}} \right)$
Divergenz: $\operatorname{div} \vec u = \nabla \cdot \vec u = \frac{{\partial u}}{{\partial x}}+\frac{{\partial v}}{{\partial y}}+\frac{{\partial w}}{{\partial z}}$
${u_i}\frac{{\partial {u_j}}}{{\partial {x_i}}} = \left( {\vec u \cdot \nabla } \right)\vec u = \vec \omega \times \vec u+\nabla \left( {\frac{1}{2}{{\vec u}^2}} \right)\quad ;\quad \vec \omega = \nabla \times \vec u$
$\frac{{\partial {t_{ij}}}}{{\partial {x_i}\partial {x_j}}} = \nabla \cdot \left( {\nabla \cdot \underline{\underline T} } \right)$
Kleine Störungen:

$p = {p_0}+{p^\prime }$

$\rho = {\rho _0}+{\rho ^\prime }$

$\vec u = {{\vec u}_0}+{{\vec u}^\prime } = {{\vec u}^\prime }\qquad |{{\vec u}_0} = 0$
Einheiten:

$\left[ F \right] = N = \frac{{{\text{kg}}\cdot {\text{m}}}}{{{{\text{s}}^2}}}$

$J = \frac{{{\text{kg}} \cdot {{\text{m}}^{\text{2}}}}}{{{{\text{s}}^{\text{2}}}}} = V \cdot A \cdot s = C \cdot V = W \cdot s$

$\left[ P \right] = W = V \cdot A = \frac{J}{C} \cdot \frac{C}{s} = \frac{J}{s}$

$\left[ R \right] = \frac{J}{{kg\;K}}$

$\left[ \nu \right] = \frac{{{m^2}}}{s}$

$\left[ {\operatorname{Re} } \right] = \left[ {} \right]$
$\int {{{\sin }^2}\left( {ax} \right)dx = \frac{1}{2}x-\frac{1}{{4a}}\sin \left( {2ax} \right)} \quad ,\quad \int {{{\cos }^2}\left( {ax} \right)dx = \frac{1}{2}x+\frac{1}{{4a}}\cos \left( {2ax} \right)}$

1. Akustik

Kontinuitätsgleichung (Masse):

$\frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {\rho u} \right) + \frac{\partial }{{\partial y}}\left( {\rho v} \right) + \frac{\partial }{{\partial z}}\left( {\rho w} \right) = 0$

$\quad \Rightarrow \quad \boxed{\frac{{\partial \rho }}{{\partial t}} + \operatorname{div} \left( {\rho \vec u} \right) = 0}$

$\Rightarrow \quad \frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {x_i}}}\left( {\rho {u_i}} \right) = 0$

Linearisiert:

$\Rightarrow \quad \frac{{\partial {\rho ^\prime }}}{{\partial t}} + {\rho _0}\operatorname{div} {\vec u^\prime } = 0$
Navier-Stokes-Gleichungen (Impulserhaltung):

$\begin{gathered} \rho \left( {\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}}} \right) = - \frac{{\partial p}}{{\partial x}} + \eta \left( {\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} + \frac{{{\partial ^2}u}}{{\partial {z^2}}}} \right) + {k_x} \hfill \\ \rho \left( {\frac{{\partial v}}{{\partial t}} + u\frac{{\partial v}}{{\partial x}} + v\frac{{\partial v}}{{\partial y}} + w\frac{{\partial v}}{{\partial z}}} \right) = - \frac{{\partial p}}{{\partial y}} + \eta \left( {\frac{{{\partial ^2}v}}{{\partial {x^2}}} + \frac{{{\partial ^2}v}}{{\partial {y^2}}} + \frac{{{\partial ^2}v}}{{\partial {z^2}}}} \right) + {k_y} \hfill \\ \rho \left( {\frac{{\partial w}}{{\partial t}} + u\frac{{\partial w}}{{\partial x}} + v\frac{{\partial w}}{{\partial y}} + w\frac{{\partial w}}{{\partial z}}} \right) = - \frac{{\partial p}}{{\partial z}} + \eta \left( {\frac{{{\partial ^2}w}}{{\partial {x^2}}} + \frac{{{\partial ^2}w}}{{\partial {y^2}}} + \frac{{{\partial ^2}w}}{{\partial {z^2}}}} \right) + {k_z} \hfill \\ \end{gathered}$

Vernachlässigung von äußeren Kräften und Reibung $\Rightarrow$ Eulersche Gleichungen:

$\left. {\begin{array}{*{20}{c}} {\rho \left( {\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}}} \right) = - \frac{{\partial p}}{{\partial x}}} \\ {\rho \left( {\frac{{\partial v}}{{\partial t}} + u\frac{{\partial v}}{{\partial x}} + v\frac{{\partial v}}{{\partial y}} + w\frac{{\partial v}}{{\partial z}}} \right) = - \frac{{\partial p}}{{\partial y}}} \\ {\rho \left( {\frac{{\partial w}}{{\partial t}} + u\frac{{\partial w}}{{\partial x}} + v\frac{{\partial w}}{{\partial y}} + w\frac{{\partial w}}{{\partial z}}} \right) = - \frac{{\partial p}}{{\partial z}}} \end{array}} \right\} \Rightarrow \boxed{\frac{{\partial \vec u}}{{\partial t}} + \left( {\vec u \cdot \vec \nabla } \right)\vec u = - \frac{1}{\rho }\operatorname{grad} p}$

Linearisiert:

$\Rightarrow \quad {\rho _0}\frac{{\partial {{\vec u}^\prime }}}{{\partial t}} = - \operatorname{grad} {p^\prime }$
Druck-Dichte-Relation: ${p^\prime } = {\rho ^\prime }\frac{{dp}}{{d\rho }}\left( {{\rho _0}} \right) = {\rho ^\prime }{c^2}$
Schallgeschwindigkeit: ${c^2} = \frac{{dp}}{{d\rho }}\left( {{\rho _0}} \right)$
Wellengleichung (ebene Welle):

$\frac{1}{{{c^2}}}\frac{{{\partial ^2}{p^\prime }}}{{\partial {t^2}}} - \Delta {p^\prime } = 0\quad \Leftrightarrow \quad \frac{1}{{{c^2}}}\frac{{{\partial ^2}{p^\prime }}}{{\partial {t^2}}} - \frac{{{\partial ^2}{p^\prime }}}{{\partial {x^2}}} - \frac{{{\partial ^2}{p^\prime }}}{{\partial {y^2}}} - \frac{{{\partial ^2}{p^\prime }}}{{\partial {z^2}}} = 0$

Allgemeine Lösung: ${p^\prime }\left( {x,t} \right) = f\underbrace {\left( {x - ct} \right)}_{\xi \left( {x,t} \right)} + g\underbrace {\left( {x + ct} \right)}_{\eta \left( {x,t} \right)}$

Nach links: $f = 0,\;g \ne 0$ ; nach rechts (positiver Richtung): $f \ne 0,\;g = 0$

Zeitliche Ableitung: $\frac{1}{{{c^2}}}\frac{{{\partial ^2}{p^\prime }}}{{\partial {t^2}}} = \frac{{{d^2}f}}{{d{\xi ^2}}}\left( {x - ct} \right) + \frac{{{d^2}g}}{{d{\eta ^2}}}\left( {x + ct} \right)$

Räumliche Ableitung: $\Delta {p^\prime }\left( {x,t} \right) = \frac{{{\partial ^2}{p^\prime }}}{{\partial {x^2}}} = \frac{{{d^2}f}}{{d{\xi ^2}}}\left( {x - ct} \right) + \frac{{{d^2}g}}{{d{\eta ^2}}}\left( {x + ct} \right)$

Dichteverteilung: ${\rho ^\prime } = \frac{{{p^\prime }\left( {x,t} \right)}}{{{c^2}}} = \frac{1}{{{c^2}}}\left[ {f\left( {x - ct} \right) + g\left( {x + ct} \right)} \right]$
Schnelle: ${u^\prime }\left( {x,t} \right) = \frac{1}{{{\rho _0}c}}\left[ {f\left( {x - ct} \right) + g\left( {x + ct} \right)} \right]$

Schnelle in unterschiedlichen Medien: ${u^\prime } = \frac{{{p^\prime }}}{{{\rho _0}c}} = \frac{{{p^\prime }}}{Z}$
Wellenwiderstand / Akustische Impedanz: $Z = {\rho _0}c\quad = \left[ {\frac{{kg}}{{{m^2}s}}} \right]$
Wellenzahl: $k = \frac{\omega }{c} = \frac{{2\pi }}{\lambda }$
Harmonische Welle (der Druckstörung): ${p^\prime }\left( {x,t} \right) = A\cos \left( {\omega t - kx} \right) = \operatorname{Re} \left( {A{e^{i\left( {\omega t - kx} \right)}}} \right)$
Effektiv- / RMS-Wert einer Druckschwankung: $\boxed{{p_{rms}} = \sqrt {\left\langle {{p^\prime }{{\left( {x,t} \right)}^2}} \right\rangle } = \sqrt {\frac{1}{T}\int\limits_0^T {{p^\prime }{{\left( t \right)}^2}dt} } }$

(root mean square) bei einer harmonischen Welle: ${p_{rms}} = \sqrt {\frac{1}{2}} A$
Schalldruckpegel: $\boxed{{L_P} = 20 \cdot {{\log }_{10}}\left( {\frac{{{p_{rms}}}}{{2 \cdot {{10}^{ - 5}}Pa}}} \right)dB}$
Überlagerung harmonischer Wellen / Schwebung:

$s\left( {x,t} \right) = 2A\underbrace {\cos \left( {\frac{{{\omega _1} - {\omega _2}}}{2}t - \frac{{{k_1} - {k_2}}}{2}x} \right)}_{Einh\ddot ullende} \cdot \underbrace {\cos \left( {\frac{{{\omega _1} + {\omega _2}}}{2}t - \frac{{{k_1} + {k_2}}}{2}x} \right)}_{Grundschwingung}$
Phasengeschwindigkeit (Ausbreitungsgeschwindigkeit der Grundschwingung):

${c^{ph}} = \frac{{{\omega _1} + {\omega _2}}}{{{k_1} + {k_2}}}$
Gruppengeschwindigkeit (Ausbreitungsgeschwindigkeit der Einhüllenden):

${c^{gr}} = \frac{{{\omega _1} - {\omega _2}}}{{{k_1} - {k_2}}}\qquad \left( { = \frac{{d\omega \left( {{k_1}} \right)}}{{dk}}\quad f\ddot ur\quad {k_2} \to {{\text{k}}_1}\quad wenn\quad {\omega _1} \approx {\omega _2}} \right)$

Reflexion und Transmission

${B_1} = \frac{{{Z_2} - {Z_1}}}{{{Z_2} + {Z_1}}}{A_1} = R{A_1}$

Reflexionsfaktor: $R = \frac{{\frac{{{Z_2}}}{{{Z_1}}} - 1}}{{\frac{{{Z_2}}}{{{Z_1}}} + 1}}$ ; Transmissionsfaktor: $T = 1 + R$
Amplitude der transmittierten Welle:
${A_2} = {A_1} + {B_1} = \left( {1 + \frac{{{Z_2} - {Z_1}}}{{{Z_2} + {Z_1}}}} \right){A_1} = \left( {1 + R} \right){A_1} = T{A_1}$
Brechung: $\frac{{\sin \alpha }}{{\sin \beta }} = \frac{{{c_1}}}{{{c_2}}}$
Schallausbreitung bei Strömung:

${x_R} = {x_B} + Ut$

$p_R^\prime \left( {{x_R},t} \right) = p_R^\prime \left( {{x_B} + Ut,t} \right) = p_B^\prime \left( {{x_B},t} \right)$

${\omega _R} = {\omega _B} + kU = {\omega _B} + \frac{{{\omega _B}}}{c}U = {\omega _B}\underbrace {\left( {1 + Ma} \right)}_{Dopplerfaktor}$

Martinshorn: $\boxed{{\lambda _b} = {\lambda _a} - \frac{u}{f} = \frac{c}{f} - \frac{u}{f}\quad ;\quad {f_b} = \frac{c}{{{\lambda _b}}}}$ (annäherndes Fahrzeug)
Machwinkel: $\boxed{\sin \mu = \frac{{ct}}{{ut}} = \frac{c}{u} = \frac{1}{{Ma}}\quad \Rightarrow \quad \mu = \arcsin \left( {\frac{1}{{Ma}}} \right)}$

Inhomogene Wellengleichung

Konti+Reibungsbehaftete Navier-Stokes-Gleichung:

$\Rightarrow \quad \frac{\partial }{{\partial t}}\left( {\rho {u_i}} \right) + \frac{\partial }{{\partial {x_j}}}\left( {\rho {u_i}{u_j}} \right) = - \frac{{\partial p}}{{\partial {x_i}}} + \frac{{\partial {\tau _{ij}}}}{{\partial {x_j}}}$

${P_{ij}} = \left( {p - {p_0}} \right){\delta _{ij}} - {\tau _{ij}} = {p^\prime }{\delta _{ij}} - {\tau _{ij}} = - \left( {\begin{array}{*{20}{c}} {{\tau _{11}} - {p^\prime }}&{{\tau _{12}}}&{{\tau _{13}}} \\ {{\tau _{21}}}&{{\tau _{22}} - {p^\prime }}&{{\tau _{23}}} \\ {{\tau _{31}}}&{{\tau _{32}}}&{{\tau _{33}} - {p^\prime }} \end{array}} \right)$

${\delta _{ij}} = \left\{ {\begin{array}{*{20}{c}} {1\quad f\ddot ur\quad i = j} \\ {0\quad f\ddot ur\quad i \ne j} \end{array}} \right.$

$\Rightarrow \quad \frac{\partial }{{\partial t}}\left( {\rho {u_i}} \right) + \frac{\partial }{{\partial {x_j}}}\left( {\rho {u_i}{u_j} + {P_{ij}}} \right) = 0$

Lighthillscher Spannungstensor: ${T_{ij}} = \rho {u_i}{u_j} + {P_{ij}} - c_0^2{\delta _{ij}}{\rho ^\prime } = \rho {u_i}{u_j} - {\tau _{ij}} + {\delta _{ij}}\left( {{p^\prime } - c_0^2{\rho ^\prime }} \right)$

$\Rightarrow \quad \frac{{{\partial ^2}{\rho ^\prime }}}{{\partial {t^2}}} - c_0^2\Delta {\rho ^\prime } = \frac{{{\partial ^2}{T_{ij}}}}{{\partial {x_i}\partial {x_j}}}$ (Lighthill-Gleichung)
Newtonscher Reibungsansatz: ${\tau _{ij}} = \eta \left( {\frac{{\partial {u_j}}}{{\partial {x_i}}} + \frac{{\partial {u_i}}}{{\partial {x_j}}}} \right)$
Abschätzungen: ${\left| {{T_{ij}}} \right|_{\max }} \ll {\left| {c_0^2\Delta {\rho ^\prime }} \right|_{\max }}\quad ,\quad {\left| {\frac{{{\partial ^2}{T_{ij}}}}{{\partial {x_i}\partial {x_j}}}} \right|_{\max }} \ll {\left| {c_0^2\Delta {\rho ^\prime }} \right|_{\max }}$
Reynoldsscher Spannungstensor: ${\tau _{tur}} = \left( { - \rho } \right) \cdot \left( {\begin{array}{*{20}{c}} {\overline {{u^{\prime 2}}} }&{\overline {{u^\prime }{v^\prime }} }&{\overline {{u^\prime }{w^\prime }} } \\ {\overline {{v^\prime }{u^\prime }} }&{\overline {{v^{\prime 2}}} }&{\overline {{v^\prime }{w^\prime }} } \\ {\overline {{w^\prime }{u^\prime }} }&{\overline {{w^\prime }{v^\prime }} }&{\overline {{w^{\prime 2}}} } \end{array}} \right)$
Lösung der Lighthill-Gleichung (Schallfeld im Fernfeld):

${\rho ^\prime }\left( {\vec x,t} \right) = \frac{1}{{4\pi c_0^2}}\frac{{{\partial ^2}}}{{\partial {x_i}\partial {x_j}}}\int\limits_{{V_Q}} {\frac{{{T_{ij}}\left( {\vec y,t - \left| {\vec x - \vec y} \right|/{c_0}} \right)}}{{\left| {\vec x - \vec y} \right|}}{d^3}y}$ (V_Q: Begrenztes Volumen)

Schalldruckfeld einer oszillierenden Kugel (Kugelkoord.):
${p^\prime } = - {\rho _0}\frac{{\partial \phi }}{{\partial t}}$
Ma⁸-Gesetz (mittlere Schalleistung):

$\left\langle {{\rho ^\prime }} \right\rangle \propto \rho _0^2M{a^8}\frac{{{D^2}}}{{{R^2}}}$

$\Delta t \propto \frac{D}{U}\quad \Rightarrow \quad \lambda = {c_0}\Delta t \propto {c_0}\frac{D}{U}\quad \Rightarrow \quad \lambda \propto \frac{D}{{Ma}}$
Lärmerzeugung (Dipolverteilung, Impulszufuhr, oszillierende Wirbel, Wirbelschall):

$\frac{{{\partial ^2}{\rho ^\prime }}}{{\partial {t^2}}} - c_0^2\Delta {\rho ^\prime } = {\rho _0}\operatorname{div} \left( {\vec \omega \times \vec u} \right)$

2. Thermodynamische Grundlagen

Zustandsgleichungen:
Virialentwicklung: $\frac{{p{V_m}}}{{{R_m}T}} = Z = 1 + {B^\prime }\frac{p}{{{R_m}T}} + {C^\prime }\frac{{{p^2}}}{{{R_m}T}} + \ldots$ ; Z: Realgasfaktor (1 $\Rightarrow$ id. Gas.)
Van-der-Waals-Gleichung: $\left( {p + \frac{a}{{V_m^2}}} \right)\left( {{V_m} - b} \right) = {R_m}T$
(a: Kohäsionsdruck; b: Kovolumen; Z = 1 $\Rightarrow$ a = b = 0)
Ideale Gasgleichung:

$\boxed{p{V_m} = {R_m}T\quad \Leftrightarrow \quad pV = mRT\quad \Leftrightarrow \quad \frac{p}{\rho } = RT\quad \Leftrightarrow \quad p\nu = RT}$

$mRT = m\frac{{{R_m}}}{M}T = n{R_m}T = n{N_A}{k_B}T = N{k_B}T$

$R = \frac{{{R_m}}}{M}$ in $\frac{J}{{kg\;K}}$ : spezifische oder spezielle Gaskonstante

${R_m} = {N_A}{k_B}$ in $\frac{J}{{mol\;K}}$ : universelle / molare Gaskonstante $8,314472 \times {10^{23}}mo{l^{ - 1}}$

$m = N\;{m_T}$ in $kg$ : Masse ( ${m_T}$ ist die Masse einer Molekel in kg)

$n = \frac{N}{{{N_A}}}$ in $\frac{{kg}}{{mol}}$ : Stoffmenge

$M = \frac{m}{n} = {N_A}{m_T}$ in $\frac{{kg}}{{mol}}$ : molare Masse oder Molmasse

$N$ : Teilchenzahl

${N_A}$ in $\frac{1}{{mol}}$ : Avogadro-Konstante $6,02214179 \times {10^{23}}mo{l^{ - 1}}$

${k_B}$ in $\frac{J}{K}$ : Boltzmann-Konstante $1,3806504\left( {24} \right) \times {10^{ - 23}}\frac{J}{K}$
Kalorische Zustandsgleichung oder Energiegleichung

Innere Energie: ${u^*} = {u^*}\left( {T,\rho } \right) = {E_{ges}} - {E_{kin}} - {E_{pot}}$

Als totales Differential: $d{u^*} = {\left( {\frac{{\partial {u^*}}}{{\partial T}}} \right)_V}dT + {\left( {\frac{{\partial {u^*}}}{{\partial \rho }}} \right)_T}d\rho$

Für ein thermisch und kalorisch ideales Gas: ${u^*} = {c_V}T + konst.$

spez. Wärmekapazität bei konst. Volumen: ${c_V} = {\left( {\frac{{\partial {u^*}}}{{\partial T}}} \right)_V}$ (= konst. bei idealem Gas)
Kalorische Zustandsgleichung mit Volumenänderungsarbeit

Enthalpie: $h = {u^*} + pV$

Als totales Differential: $dh = {\left( {\frac{{\partial h}}{{\partial T}}} \right)_p}dT + {\left( {\frac{{\partial h}}{{\partial p}}} \right)_T}dp$

Für ein thermisch und kalorisch ideales Gas: $h = {c_p}T + konst. = {u^*} + pV = {u^*} + RT = h\left( T \right)$

spez. Wärmekapazität bei konst. Druck: ${c_p} = {\left( {\frac{{\partial h}}{{\partial T}}} \right)_p}$ (= konst. bei idealem Gas)
Kalorische Zustandsgleichung mit Volumenarbeit und Wärmeaustausch

Entropie: $s = s\left( {T,p} \right)$

Als totales Differential: $ds = {\left( {\frac{{\partial s}}{{\partial T}}} \right)_p}dT + {\left( {\frac{{\partial s}}{{\partial p}}} \right)_T}dp$

Für ein thermisch und kalorisch ideales Gas:

Fundamentalgleichung: $Tds = d{u^*} + pdV = dh - Vdp$ $\quad \Leftrightarrow \quad ds = \frac{1}{T}\left( {d{u^*} + pdV} \right)$

$\left( {h = {u^*} + pV\quad \Rightarrow \quad dh = d{u^*} + pdV + Vdp} \right)$

$\begin{gathered} s = {s_2} - {s_1} = {c_V}\ln \frac{{{T_2}}}{{{T_1}}} + R\ln \frac{{{V_2}}}{{{V_1}}} \hfill \\ s = {s_2} - {s_1} = {c_p}\ln \frac{{{T_2}}}{{{T_1}}} - R\ln \frac{{{p_2}}}{{{p_1}}} \hfill \\ \end{gathered}$
1. Hauptsatz der Thermodynamik: $Q + W = \Delta {u^*}$
2. Hauptsatz der Thermodynamik: $\Delta s \geqslant 0$
$\kappa = \frac{{{c_p}}}{{{c_V}}}\quad \left( { = 1,4\;bei\;Luft} \right),\quad {c_p} - {c_V} = R$
isotherm: $\Delta s = R\ln \frac{{{V_2}}}{{{V_1}}}$ ; isochor: $\Delta s = {c_V}ln\frac{{{T_2}}}{{{T_1}}}$
isobar: $\Delta s = {c_p}ln\frac{{{T_2}}}{{{T_1}}}$ ; adiabat: $\Delta s = 0$
Bei isentropen (adiabat, reversibel) Zustandsänderungen gilt: $\boxed{p \sim {\rho ^\kappa } \sim {T^{\frac{\kappa }{{\kappa - 1}}}}}$

3. Eindimensionale Gasströmungen

Erhaltungsgleichungen

Massenerhaltung: $\rho uA = konst\quad \Rightarrow \quad d\left( {\rho uA} \right) = 0$

Impulserhaltung: $\frac{{{u^2}}}{2} + \int {\frac{{dp}}{\rho }} = konst\quad ,\quad \rho \;u\;du + dp = 0$

Energieerhaltung: $h + \frac{{{u^2}}}{2} = konst = {h_0}\quad \Rightarrow \quad dh + d\left( {\frac{{{u^2}}}{2}} \right) = 0$
Energiegleichung:

Für ideale Gase bei Vernachlässigung der potentiellen Energie: ${c_p}T + \frac{{{u^2}}}{2} = C = konst.$

Ausströmen aus einem Kessel (stationär)

Geschwindigkeitsverlauf (aus Energiesatz): $\boxed{{c_p}T + \frac{{{u^2}}}{2} = {c_p}{T_0}}$
Ausströmgeschwindigkeit: $\boxed{u = \sqrt {2{c_p}\left( {{T_0} - T} \right)} }$ ; ${u_{\max }} = \sqrt {2{c_p}{T_0}}$
$\frac{u}{{{u_{\max }}}} = \frac{{\sqrt {2{c_p}\left( {{T_0} - T} \right)} }}{{\sqrt {2{c_p}{T_0}} }} = \sqrt {1 - \frac{T}{{{T_0}}}} = \sqrt {1 - {{\left( {\frac{p}{{{p_0}}}} \right)}^{\frac{{\kappa - 1}}{\kappa }}}}$ mit $p \sim {\rho ^\kappa } \sim {T^{\frac{\kappa }{{\kappa - 1}}}}$
Dichteverlauf (aus Isentrope): $\boxed{\frac{\rho }{{{\rho _0}}} = {{\left( {\frac{p}{{{p_0}}}} \right)}^{\frac{1}{\kappa }}}}$
Temperaturverlauf (aus Zustandsgleichung und Isentrope): $\boxed{\frac{T}{{{T_0}}} = {{\left( {\frac{p}{{{p_0}}}} \right)}^{\frac{{\kappa - 1}}{\kappa }}}}$
Machzahlverlauf: $\boxed{Ma = \frac{u}{c} = \sqrt {\frac{2}{{\kappa - 1}}\left[ {{{\left( {\frac{{{p_0}}}{p}} \right)}^{\frac{{\kappa - 1}}{\kappa }}} - 1} \right]} }$
Kritische Werte (für $Ma = 1$ ) (bei $\kappa = 1,4$ ):

$\frac{{{u^*}}}{{{u_{\max }}}} = \sqrt {\frac{{\kappa - 1}}{{\kappa + 1}}} = 0,408$

$\frac{{{T^*}}}{{{T_0}}} = \frac{2}{{\kappa + 1}} = 0,833$

$\frac{{{p^*}}}{{{p_0}}} = {\left( {\frac{2}{{\kappa + 1}}} \right)^{\frac{\kappa }{{\kappa - 1}}}} = 0,528$ $\frac{{{\rho ^*}}}{{{\rho _0}}} = {\left( {\frac{2}{{\kappa + 1}}} \right)^{\frac{1}{{\kappa - 1}}}} = 0,634$

$M{a^*} = \frac{u}{c}$

${\left( {\frac{{M{a^*}}}{{Ma}}} \right)^2} = {\left( {\frac{c}{{{c^*}}}} \right)^2} = {\left( {\frac{c}{{{c_0}}} \cdot \frac{{{c_0}}}{{{c^*}}}} \right)^2} = \frac{{\frac{{\kappa + 1}}{2}}}{{1 + \frac{{\kappa - 1}}{2}M{a^2}}}\quad \Rightarrow \quad M{a^{*2}} = \frac{{\frac{{\kappa + 1}}{2}M{a^2}}}{{1 + \frac{{\kappa - 1}}{2}M{a^2}}}$

Laval-Düsenströmung

Fundamentale Beziehung: $\underbrace {\frac{{d\rho }}{\rho }}_{rel.\;Dichte\ddot anderung} = - M{a^2} \cdot \underbrace {\frac{{du}}{u}}_{rel.\;Geschw.\ddot anderung}$
$\frac{{du}}{u} = - \frac{1}{{1 - M{a^2}}} \cdot \frac{{dA}}{A} = \frac{1}{{M{a^2} - 1}} \cdot \frac{{dA}}{A}$
Verhalten:

$\begin{gathered} Ma \to 0:\quad \rho \quad \cdot \quad u \uparrow \quad \cdot \quad A \downarrow \quad = \quad konst. \hfill \\ Ma < 1:\quad \rho \downarrow \quad \cdot \quad u \uparrow \uparrow \cdot \quad A \downarrow \quad = \quad konst. \hfill \\ Ma = 1:\quad \rho \downarrow \quad \cdot \quad u \uparrow \quad \cdot \quad A\quad = \quad konst. \hfill \\ Ma > 1:\quad \rho \downarrow \downarrow \cdot \quad u \uparrow \quad \cdot \quad A \uparrow \quad = \quad konst. \hfill \\ \end{gathered}$
Stromdichte (Maß für den Platzbedarf einer Strömung): $\rho u$
$\frac{{d\left( {\rho u} \right)}}{{\rho u}} + \frac{{dA}}{A} = 0\quad \Rightarrow \quad \frac{{d\left( {\rho u} \right)}}{{\rho u}} = \left( {1 - M{a^2}} \right) \cdot \frac{{du}}{u}$ ; ${A_{\min }}\quad \leftrightarrow \quad \rho u = {\left( {\rho u} \right)_{\max }}$
DGL für Ma(x) bei gegebenem A(x): $\frac{1}{{Ma}} \cdot \frac{{dMa}}{{dx}} = - \frac{{1 + \frac{{\kappa - 1}}{2}M{a^2}}}{{1 - M{a^2}}} \cdot \frac{1}{A}\frac{{dA}}{{dx}}$
Diagramm F
- $\boxed{\frac{{{p_1}}}{{{p_0}}} = \frac{1}{{{{\left( {1 + \frac{{\kappa - 1}}{2}Ma_1^2} \right)}^{\frac{\kappa }{{\kappa - 1}}}}}}}$
- $\boxed{\frac{{{A_1}}}{{{A^*}}} = \frac{1}{{M{a_1}}}{{\left[ {\frac{2}{{\kappa + 1}}\left( {1 + \frac{{\kappa - 1}}{2}Ma_1^2} \right)} \right]}^{\frac{{\kappa + 1}}{{2\left( {\kappa - 1} \right)}}}}}$
- $\boxed{\frac{{{T_0}}}{T} = 1 + \frac{{\kappa - 1}}{2}M{a^2}}$

Kompressible Rohströmung mit Reibung

$T\frac{{ds}}{{dx}} + \nu \frac{{dp}}{{dx}} + {j^2}\nu \frac{{d\nu }}{{dx}} = \boxed{T\frac{{ds}}{{dx}}} + \boxed{\nu \frac{{dp}}{{dx}}} + \boxed{{j^2}\nu {{\left( {\frac{{\partial \nu }}{{\partial p}}} \right)}_s}\frac{{dp}}{{dx}}} + \boxed{{j^2}\nu {{\left( {\frac{{\partial \nu }}{{\partial s}}} \right)}_p}\frac{{ds}}{{dx}}} = 0$
$\frac{{dp}}{{dx}} < 0\quad ,\quad \frac{{du}}{{dx}} > 0\quad f\ddot ur\quad u < c$
$\frac{{dp}}{{dx}} > 0\quad ,\quad \frac{{du}}{{dx}} < 0\quad f\ddot ur\quad u > c$
$\boxed{\frac{{{u_1}}}{{{u_2}}} = \sqrt {1 - \frac{{2\lambda \frac{L}{d}\kappa Ma_1^2}}{{2 + \left( {\kappa - 1} \right)Ma_1^2}}} }$ ( $\lambda$ aus Moody-Diagramm) $\boxed{\operatorname{Re} = \frac{{u \cdot d}}{\nu }}$
$\boxed{\frac{{{T_2}}}{{{T_1}}} = 1 + \frac{{\kappa - 1}}{2}Ma_1^2\left[ {1 - {{\left( {\frac{{{u_2}}}{{{u_1}}}} \right)}^2}} \right]}$
(Massenbilanz, Zustandsgleichung) (s.u.)

Unstetigkeitsbedingungen

Massenstrom: ${\rho _1}{u_{1x}} = {\rho _2}{u_{2x}}$
Energiestrom: ${\rho _1}{u_{1x}}\left( {{h_1} + \frac{{u_1^2}}{2}} \right) = {\rho _2}{u_{2x}}\left( {{h_2} + \frac{{u_2^2}}{2}} \right)$
Impulsstrom:

$\begin{gathered} {\rho _1}u_{1x}^2 + {p_1} = {\rho _2}u_{2x}^2 + {p_2} \hfill \\ {\rho _1}{u_{1x}}{u_{1z}} = {\rho _2}{u_{2x}}{u_{2z}} \hfill \\ {\rho _1}{u_{1x}}{u_{1y}} = {\rho _2}{u_{2x}}{u_{2y}} \hfill \\ \end{gathered}$

Verdichtungsstoß

Massenbilanz: $\boxed{{\rho _1}{u_1} = {\rho _2}{u_2}}$
Impulsbilanz: ${\rho _1}u_1^2 + {p_1} = {\rho _2}u_2^2 + {p_2}$
Energiebilanz: ${c_p}{T_1} + \frac{{u_1^2}}{2} = {c_p}{T_2} + \frac{{u_2^2}}{2}$
Zustandsgleichung: $\boxed{\frac{{{\rho _1}{T_1}}}{{{p_1}}} = \frac{{{\rho _2}{T_2}}}{{{p_2}}}}$
$c = \sqrt {\kappa RT} = \sqrt {\kappa \frac{p}{\rho }} \quad ;\quad Ma = \frac{u}{c}\quad$
$\quad {c_p}T = {c_p}\frac{p}{{\rho R}} = \frac{{{c_p}}}{{{c_p} - {c_V}}}\frac{p}{\rho } = \frac{\kappa }{{\kappa - 1}}\frac{p}{\rho }\quad \Rightarrow \quad \boxed{{c_p} = \frac{\kappa }{{\kappa - 1}}R}$

Senkrechter Verdichtungsstoß (Normal Shock)

Stoßlösung:

$\boxed{\frac{{{p_2}}}{{{p_1}}} = 1 + \frac{{2\kappa }}{{\kappa + 1}}\left( {Ma_1^2 - 1} \right)}$

$\boxed{\frac{{{u_2}}}{{{u_1}}} = \frac{{{\rho _1}}}{{{\rho _2}}} = 1 - \frac{2}{{\kappa + 1}}\left( {1 - \frac{1}{{Ma_1^2}}} \right)}$

$\boxed{\frac{{{T_2}}}{{{T_1}}} = \frac{{{p_2}}}{{{p_1}}} \cdot \frac{{{\rho _1}}}{{{\rho _2}}} = \left[ {1 + \frac{{2\kappa }}{{\kappa + 1}}\left( {Ma_1^2 - 1} \right)} \right]\left( {\frac{{\kappa - 1}}{{\kappa + 1}} + \frac{2}{{\kappa + 1}}\frac{1}{{Ma_1^2}}} \right)}$

$\boxed{\frac{{M{a_2}}}{{M{a_1}}} = \sqrt {\frac{{\frac{2}{{Ma_1^2}} + \left( {\kappa - 1} \right)}}{{2\kappa Ma_1^2 - \left( {\kappa - 1} \right)}}} }$
Entropiezunahme (Nach dem 2. HS der Thermodynamik):

$\boxed{\frac{{{s_2} - {s_1}}}{{{c_V}}} = \ln \left[ {\left( {1 + \frac{{2\kappa }}{{\kappa + 1}}\left( {Ma_1^2 - 1} \right)} \right){{\left( {1 - \frac{2}{{\kappa + 1}}\left( {1 - \frac{1}{{Ma_1^2}}} \right)} \right)}^\kappa }} \right]}$ $\boxed{{c_V} = \frac{R}{{\kappa - 1}}}$

$Ma$ : Machzahlen, senkrecht zu den Stößen!

Starker Verdichtungsstoß:

Hyperschalllimes:

$\begin{gathered} \frac{{{\rho _2}}}{{{\rho _1}}} = \frac{{{u_1}}}{{{u_2}}}\quad \to \quad \frac{{\kappa + 1}}{{\kappa - 1}}\quad \left( { = 6} \right) \hfill \\ \frac{{{p_2}}}{{{p_1}}}\quad \to \quad \frac{{2\kappa }}{{\kappa + 1}}Ma_1^2 \sim Ma_1^2 \hfill \\ \frac{{{T_2}}}{{{T_1}}}\quad \to \quad \frac{{2\kappa \left( {\kappa - 1} \right)}}{{{{\left( {\kappa + 1} \right)}^2}}}Ma_1^2 \sim Ma_1^2 \hfill \\ \end{gathered}$

Prandtl’sche Stoßrelation

${u_2}{u_1} = {c^{*2}}\quad ;\quad Ma_2^*Ma_1^* = 1$
$Ma_1^* > 1\quad \Rightarrow \quad M{a_1} > 1,\quad Ma_2^* < 1,\quad M{a_2} < 1,\quad {u_2} < {u_1},\quad {p_2} > {p_1},\quad {\rho _2} > {\rho _1}$
Kritische Mach-Zahl: $M{a^*} = \frac{u}{{{c^*}}}$
${\left( {\frac{{M{a^*}}}{{Ma}}} \right)^2} = {\left( {\frac{c}{{{c^*}}}} \right)^2} = {\left( {\frac{c}{{{c_0}}} \cdot \frac{{{c_0}}}{{{c^*}}}} \right)^2} = \frac{{\frac{{\kappa + 1}}{2}}}{{1 + \frac{{\kappa - 1}}{2}M{a^2}}}\quad \Rightarrow \quad \boxed{M{a^*} = \sqrt {\frac{{\frac{{\kappa + 1}}{2}M{a^2}}}{{1 + \frac{{\kappa - 1}}{2}M{a^2}}}} }$ (Stoßpolarendiagramm)

4. Einfache Sonderfälle zweidimensionaler Strömungen

Schiefer Verdichtungsstoß (Mach Wave)

Stoßlösung:

$\frac{{{p_2} - {p_1}}}{{{p_1}}} = \frac{{2\kappa }}{{\kappa + 1}}\left[ {{{\left( {M{a_1}\sin \beta } \right)}^2} - 1} \right]\quad ;\quad \frac{{{u_2}}}{{{u_1}}}\left( { = \frac{{{\rho _1}}}{{{\rho _2}}}} \right) = \frac{{\kappa - 1}}{{\kappa + 1}} + \frac{2}{{\kappa + 1}} \cdot \frac{1}{{{{\left( {M{a_1}\sin \beta } \right)}^2}}}$

$\boxed{\frac{{{T_2}}}{{{T_1}}}\left( { = \frac{{{p_2}}}{{{p_1}}} \cdot \frac{{{\rho _1}}}{{{\rho _2}}}} \right) = \underbrace {\left[ {1 + \frac{{2\kappa }}{{\kappa + 1}}\left\{ {{{\left( {M{a_1}\sin \beta } \right)}^2} - 1} \right\}} \right]}_{{p_2}/{p_1}} \cdot \left( {\frac{{\kappa - 1}}{{\kappa + 1}} + \frac{2}{{\kappa + 1}}\frac{1}{{{{\left( {M{a_1}\sin \beta } \right)}^2}}}} \right)}$

${p_2}/{p_1}$ : Diagramm (Initial Mach Number / Pressure Ratio)

$\boxed{\frac{{{p_2}}}{{{p_1}}} = \frac{{{p_2}}}{{{p_0}}} \cdot \frac{{{p_0}}}{{{p_1}}}}$
Bereich möglicher Stoßwinkel (wave angle degrees):

$\arcsin \frac{1}{{M{a_1}}} \leqslant \beta \leqslant \frac{\pi }{2}$

${\beta _{\max }}$ : senkrechter Stoß, größter Druckanstieg

${\beta _{\min }}$ : Mach-Linie, Stoß verschwindender Stärke
Mach-Zahl hinter dem Stoß:

$\boxed{Ma_2^2{{\sin }^2}\left( {\beta - \vartheta } \right) = \frac{{1 + \left( {\frac{{\kappa - 1}}{2}} \right)Ma_1^2{{\sin }^2}\beta }}{{\kappa Ma_1^2{{\sin }^2}\beta - \left( {\frac{{\kappa - 1}}{2}} \right)}}}$ (Diagramm: [B] {Initial- / Final Mach Number})

Ablenkwinkel (deflection angle): $\vartheta$ (evtl. $\alpha /2$ )
Umlenk- / Ablenkungswinkel (Diagramm: {Initial Mach Number / Wave Angle Degree}):

$\boxed{\tan \vartheta = 2\cot \beta \frac{{{{\left( {M{a_1}\sin \beta } \right)}^2} - 1}}{{Ma_1^2\left( {\kappa + \cos 2\beta } \right) + 2}}}$

Für große Machzahlen (Hypersonisch): $\beta = \frac{{\kappa + 1}}{2}\vartheta \sim \vartheta$
Stoßpolarendiagramm von Busemann:

$v_2^2 = {\left( {{u_1} - {u_2}} \right)^2}\frac{{{u_1}{u_2} - {c^{*2}}}}{{\frac{2}{{\kappa + 1}}u_1^2 - {u_1}{u_2} + {c^{*2}}}}$

Prandtl-Meyer-Strömung

Mach-Winkel: $\boxed{\mu = \arcsin \frac{1}{{Ma}}}$
Schiefer Stoß:

${\left( {M{a_1}\sin \beta } \right)^2} - 1 \approx \frac{{\kappa + 1}}{2}\frac{{Ma_1^2}}{{\sqrt {Ma_1^2 - 1} }}\vartheta$
$\frac{{{p_2} - {p_1}}}{{{p_1}}} = \frac{{\Delta p}}{{{p_1}}} \approx \frac{{\kappa Ma_1^2}}{{\sqrt {Ma_1^2 - 1} }}\vartheta \sim\vartheta$
Prandtl-Meyer-Funktion (Diagramm: Prandtl-Meyer Eckenströmung):

$\boxed{\nu \left( {Ma} \right) = \sqrt {\frac{{\kappa + 1}}{{\kappa - 1}}} \arctan \sqrt {\frac{{\kappa - 1}}{{\kappa + 1}}\left( {M{a^2} - 1} \right)} - \arctan \sqrt {M{a^2} - 1} = \left[ ^\circ \right]}$

Kompression: $\nu = {\nu _1} - \left| {\vartheta - {\vartheta _1}} \right|\quad \to \quad \nu = {\nu _1} - \vartheta$

Expansion: $\nu = {\nu _1} + \left| {\vartheta - {\vartheta _1}} \right|\quad \to \quad \boxed{\nu = {\nu _1} + \vartheta }$

Für gewöhnlich: ${\vartheta _1} = 0$
Größtmöglicher Umlenkwinkel (für Luft $\kappa = 1,4$ ):

${\nu _{\max }} = \nu \left( {Ma \to \infty } \right) = \frac{\pi }{2}\left( {\sqrt {\frac{{\kappa + 1}}{{\kappa - 1}}} - 1} \right) = 130.5^\circ$
Entropiezunahme:

${\left( {{s_2} - {s_1}} \right)_{Kompr.}}\sim n{\left( {\Delta \vartheta } \right)^3}\sim\vartheta {\left( {\Delta \vartheta } \right)^2}$

${\left( {{s_2} - {s_1}} \right)_{Sto{\ss}}}\sim{\vartheta ^3}\sim\vartheta {\left( {n\Delta \theta } \right)^2}\sim{n^2}{\left( {{s_2} - {s_1}} \right)_{Kompr.}}$

Stoß-Expansions-Theorie (für schlanke Profile)

Druckänderung für schlanke Profile $\left( {\vartheta \ll 1} \right)$ : $\frac{{\Delta p}}{p} = \frac{{\kappa M{a^2}}}{{\sqrt {M{a^2} - 1} }}\Delta \vartheta \sim\vartheta$
Druckkoeffizient: $\boxed{{c_p} = \frac{{2\vartheta }}{{\sqrt {M{a^2} - 1} }} = \left[ {\frac{2}{{\kappa Ma_\infty ^2}}\left( {\frac{p}{{{p_\infty }}} - 1} \right)} \right]}$
Beispiel: Doppelkeilprofil (Länge L, Dicke d)

Wellenwiderstand (pro Einheitstiefe):

${F_w} = \left( {{p_2} - {p_3}} \right)d = \left( {{p_2} - {p_3}} \right)\varepsilon L = \frac{{4{\varepsilon ^2}}}{{\sqrt {Ma_1^2 - 1} }}{q_1}L\qquad ;\qquad {q_1} = \frac{{{\rho _1}}}{2}u_1^2$

Widerstandsbeiwert (auf Einheitstiefe 1 bezogen): ${c_w} = \frac{{{F_w}}}{{{q_1}L}} = \frac{{4{\varepsilon ^2}}}{{\sqrt {Ma_1^2 - 1} }}\sim{\varepsilon ^2}\sim{\left( {\frac{d}{L}} \right)^2}$
Beispiel: Ebene angestellte Platte (Länge L, Anstellwinkel $\alpha$ )

Widerstandsbeiwert: ${c_w} = \frac{{4{\alpha ^2}}}{{\sqrt {Ma_1^2} - 1}}\sim{\alpha ^2}$

Auftriebsbeiwert: ${c_a} = \frac{{4\alpha }}{{\sqrt {Ma_1^2} - 1}}\sim\alpha \qquad ;\qquad \boxed{{c_a} = \frac{1}{l}\int\limits_0^l {\left( {{c_{p,u}} - {c_{p,o}}} \right)dx} }$
Allgemein:

Wellenwiderstand: $\boxed{{c_W} = \frac{2}{{\sqrt {Ma_\infty ^2 - 1} }}\frac{1}{c}\left( {\int\limits_0^c {{{\left( {\frac{{d{y_u}}}{{dx}}} \right)}^2}dx} + \int\limits_0^c {{{\left( {\frac{{d{y_o}}}{{dx}}} \right)}^2}dx} } \right)}$

5. Die Bewegungsgleichungen idealer Fluide

Konsequenzen aus den Bilanzgleichungen

Gesamtenthalpie: ${h_0} = h + \frac{{u_j^2}}{2} = konst.$
Entropie: $\frac{{\partial s}}{{\partial {x_k}}} = 0$
Wirbelsatz von Crocco: $\vec u \times \operatorname{rot} \vec u = \operatorname{grad} {h_0} - T\operatorname{grad} s$
Isentropenbeziehung: $\frac{p}{{{p_0}}} = {\left( {\frac{\rho }{{{\rho _0}}}} \right)^\kappa }$

Gasdynamische Grundgleichung

Gasdynamische Grundgleichung: $\left( {{u^2} - {c^2}} \right)\frac{{\partial u}}{{\partial x}} + uv\left( {\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}} \right) + \left( {{v^2} - {c^2}} \right)\frac{{\partial v}}{{\partial y}} = 0$
${h_0} = h + \frac{1}{2}\left( {{u^2} + {v^2}} \right) = {c_p}T + \frac{1}{2}\left( {{u^2} + {v^2}} \right) = \frac{\kappa }{{\kappa - 1}}\frac{p}{\rho } + \frac{1}{2}\left( {{u^2} + {v^2}} \right) = \frac{{{c^2}}}{{\kappa - 1}} + \frac{1}{2}\left( {{u^2} + {v^2}} \right) = konst.$
Drehungsfreiheit: $\frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}} = 0\qquad ;\qquad u = \frac{{\partial \Phi }}{{\partial x}},\quad v = \frac{{\partial \Phi }}{{\partial y}}$
Gasdynamische Grundgleichung: $\Rightarrow \quad \left( {1 - \frac{{\Phi _x^2}}{{{c^2}}}} \right){\Phi _{xx}} - 2\frac{{{\Phi _x}{\Phi _y}}}{{{c^2}}}{\Phi _{xy}} + \left( {1 - \frac{{\Phi _y^2}}{{{c^2}}}} \right){\Phi _{yy}} = 0$

6. Linearisierte, ebene, stationäre Strömungen

Theorie kleiner Störungen

Linearisierung: $\frac{u}{{{u_\infty }}},\frac{v}{{{u_\infty }}} \ll 1$
Linearisierte gasdynamische Grundgleichung: $\left( {1 - Ma_\infty ^2} \right){\Phi _{xx}} + {\Phi _{yy}} = 0$
Druck- und Geschwindigkeitsstörung: $\boxed{{c_p} = \frac{{p - {p_\infty }}}{{\frac{1}{2}{\rho _\infty }u_\infty ^2}} = - 2\frac{u}{{{u_\infty }}}}$
Linearisierte Potentialgleichung:

$\boxed{u = \frac{{\partial \phi }}{{\partial x}} = \frac{{\partial \phi }}{{\partial \xi }} + \frac{{\partial \phi }}{{\partial \eta }}}\qquad ;\qquad \boxed{v = \frac{{\partial \phi }}{{\partial y}} = - \lambda \frac{{\partial \phi }}{{\partial \xi }} + \lambda \frac{{\partial \phi }}{{\partial \eta }}}\qquad ;\qquad \boxed{{c_p} = - 2\frac{u}{{{u_\infty }}}}$

An der Wand: $\boxed{v = {{\left( {\frac{{\partial \phi }}{{\partial y}}} \right)}_{y = 0}} = {u_\infty }{{\left( {\frac{{dy}}{{dx}}} \right)}_{K\left( {ontur} \right)}}}$

$\boxed{\phi \left( {x,y} \right) = \underbrace {g\left( \eta \right)}_{rechtsl\"a ufig\;\left( \backslash \right)} + \underbrace {f\left( \xi \right)}_{linksl\"a ufig\;\left( / \right)} = g\left( {x + \lambda y} \right) + f\left( {x - \lambda y} \right)}$

Die Strömung längs einer welligen Wand

Linearisierte ebene Unterschallströmung

Unterschallströmung: $1 - Ma_\infty ^2 \equiv {m^2} > 0$

Potentialgleichung: ${\phi _{xx}} + \frac{1}{{{m^2}}}{\phi _{yy}} = 0$

$\begin{gathered} u = {\phi _x} = \frac{{{u_\infty }\varepsilon \alpha }}{m}\sin \alpha x \cdot \exp \left( { - m\alpha y} \right)\qquad ;\qquad v = {\phi _y} = {u_\infty }\varepsilon \alpha \cos \alpha x \cdot \exp \left( { - m\alpha y} \right) \hfill \\ {c_p} = - 2\frac{u}{{{u_\infty }}} = - 2\frac{{\varepsilon \alpha }}{m}\sin \alpha x \cdot \exp \left( { - m\alpha y} \right)\qquad ;\qquad {c_p}\left( {x,0} \right) = - 2\frac{{\varepsilon \alpha }}{m}\sin \alpha x \hfill \\ \end{gathered}$

Linearisierte ebene Überschallströmung

Überschallströmung: $Ma_\infty ^2 - 1 \equiv {\lambda ^2} > 0$

Potentialgleichung: ${\phi _{xx}} - \frac{1}{{{\lambda ^2}}}{\phi _{yy}} = 0$

$\tan \mu = \frac{1}{{\sqrt {Ma_\infty ^2 - 1} }} = \frac{1}{\lambda }$ ; Pot.Gl.: $\Rightarrow \quad \frac{{{\partial ^2}\phi }}{{\partial \xi \;\partial \eta }} = 0$

Allgemeine d’Alembertsche Lösung der Potentialgleichung:

$\begin{gathered} \phi \left( {\xi ,\eta } \right) = f\left( \xi \right) + g\left( \eta \right) \hfill \\ \phi \left( {x,y} \right) = f\left( {x - \lambda y} \right) + g\left( {x + \lambda y} \right) \hfill \\ \end{gathered}$

$\begin{gathered} u = \frac{{\partial \phi }}{{\partial x}} = - \frac{{{u_\infty }\varepsilon \alpha }}{\lambda }\cos \alpha \left( {x - \lambda y} \right)\qquad ;\qquad v = \frac{{\partial \phi }}{{\partial y}} = {u_\infty }\varepsilon \alpha \cos \alpha \left( {x - \lambda y} \right) \hfill \\ {c_p} = - 2\frac{u}{{{u_\infty }}} = 2\frac{{\varepsilon \alpha }}{\lambda }\cos \alpha \left( {x - \lambda y} \right)\qquad ;\qquad {c_p}\left( {x,0} \right) = 2\frac{{\varepsilon \alpha }}{\lambda }\cos \alpha x \hfill \\ \end{gathered}$

7. Das Charakteristikenverfahren

Das Charakteristikenverfahren

Strömungswinkel: $\tan \vartheta = \frac{v}{u}$ ; ${\left( {\frac{{dy}}{{dx}}} \right)_{\eta = konst.}} = \tan \left( {\vartheta + \mu } \right)\quad ;\quad {\left( {\frac{{dy}}{{dx}}} \right)_{\xi = konst.}} = \tan \left( {\vartheta - \mu } \right)$

Die Verträglichkeitsbedingungen

Radius der Stromlinie: R
Grundgleichungen in s,n-Koordinaten:
- Kontigleichung: $\rho w\Delta n = konst.$
- Kräftegl. in s-Richtung (Stromlinie): $\rho w\frac{{\partial w}}{{\partial s}} = - \frac{{\partial p}}{{\partial s}}$
- Kräftegl. in n-Richtung (Normale): $\rho \frac{{{w^2}}}{R} = - \frac{{\partial p}}{{\partial n}} = \rho {w^2}\frac{{\partial \vartheta }}{{\partial s}}$
- Energiegleichung: ${h_0} = h + \frac{{{w^2}}}{2} = konst.$
$w \cdot r = konst.$

$\begin{gathered} \frac{{dv}}{{ds}} - \tan \mu \frac{{d\vartheta }}{{dn}} = 0 \hfill \\ \tan \mu \frac{{dv}}{{dn}} - \frac{{d\vartheta }}{{ds}} = 0 \hfill \\ \end{gathered}$
Verträglichkeitsbedingungen (zwischen $v$ und $\vartheta$ ):

$\frac{\partial }{{\partial n}}\left( {v - \vartheta } \right) = 0\quad \Rightarrow \quad v - \vartheta = R\left( \xi \right)$ (konst. entlang $\eta$ Charakteristiken)

$\frac{\partial }{{\partial \xi }}\left( {v + \vartheta } \right) = 0\quad \Rightarrow \quad v + \vartheta = Q\left( \eta \right)$ (konst. entlang $\xi$ Charakteristiken)

$\Rightarrow \quad v = \frac{1}{2}\left( {Q + R} \right)\quad ;\quad \vartheta = \frac{1}{2}\left( {Q - R} \right)$

Riemann’sche Invarianten: $Q\left( \eta \right),\;R\left( \xi \right)$

Schema:
$\begin{array}{*{20}{c}} {{v_3}}&\vline & {{\vartheta _3}} \\ \hline {{Q_1}}&\vline & {{R_2}} \end{array}$

8. Hyperschall-Versuchsanlagen

9. Grenzschichten in kompressiblen Strömungen

Prandtlzahl

Machzahl: $Ma = \frac{u}{c}$ (Strömungs- / Schallgeschwindigkeit)
Reynoldszahl: $\operatorname{Re} = \frac{{\rho ul}}{\mu }$ (Trägheitskraft / Reibungskraft)
Prandtlzahl: $\Pr = \frac{{\mu {c_p}}}{\lambda } = \frac{\nu }{a}$ (Impulsdiffusion / Wärmetransport)
Temperaturleitfähigkeit: $a = \frac{\lambda }{{\rho {c_p}}}$ (thermal diffusity) (Wärmeleitfähigkeit /
Volumenbezogene Wärmekapazität)
kinematische Viskosität: $\nu = \frac{\mu }{\rho }$ (momentum diffusity)
Dicke von Temperaturgrenzschicht ${\delta _T}$ zu Strömungsgrenzschicht ${\delta _S}$ : $\frac{{{\delta _T}}}{{{\delta _S}}}\sim\frac{1}{{\sqrt {\Pr } }}$

Dissipation und Recovery-Faktor

Recovery-Faktor: $r = \frac{{{T_{aw}} - {T_\infty }}}{{{T_0} - {T_\infty }}} = \frac{{{T_{aw}} - {T_\infty }}}{{\frac{{u_\infty ^2}}{{2{c_p}}}}}$

Verdrängungswirkung der laminaren kompressiblen Grenzschicht

Verdrängungsdicke: ${\delta ^*}$
Grenzschichtdicke: $\delta$
${\delta ^*} = \int\limits_0^\delta {\left( {1 - \frac{{\rho u}}{{{\rho _\infty }{u_\infty }}}} \right)dy} \quad \to \quad \int\limits_0^\delta {dy} = \delta$
$\frac{\delta }{l}\sim\frac{{{\delta ^*}}}{l} = \frac{{2{\nu _\infty }}}{{k\sqrt {{{\operatorname{Re} }_\infty }} }}\left( {1 + \frac{{\kappa - 1}}{2}Ma_\infty ^2} \right)$

Wandschubspannungen bei kompressiblen Grenzschichtströmungen

Potenzsatz: $\frac{\mu }{{{\mu _\infty }}} = {\left( {\frac{T}{{{T_\infty }}}} \right)^\omega }\quad mit\quad \frac{1}{2} < \omega < 1\quad und\quad \begin{array}{*{20}{c}} {\omega = 0,5}& \to &{T \gg 0^\circ C} \\ {\omega = 0,8}& \to &{T = 0^\circ C} \\ {\omega = 1,0}& \to &{T \ll 0^\circ C} \end{array}$
$\tau = \mu \frac{{\partial u}}{{\partial y}}\quad \Rightarrow \quad \tau = {\mu _\infty }{\left( {\frac{T}{{{T_\infty }}}} \right)^\omega }\frac{{\partial u}}{{\partial y}}$

10. Turbulente und kompressible Grenzschichtströmungen

Korrelationskoeffizient: ${C_{uT}} = \frac{{\overline {{u^\prime }{T^\prime }} }}{{\sqrt {\overline {{u^{\prime 2}}} } \sqrt {\overline {{T^{\prime 2}}} } }}\qquad ;\qquad 0,6 \leqslant {C_{uT}} \leqslant 0,8$
Dichte: $\rho = \overline \rho + {\rho ^\prime }$
Ideales Gasgesetz: $\overline {{\rho ^\prime }{T^\prime }} \ll \bar \rho \bar T\quad \Rightarrow \quad \overline p = \overline \rho R\overline T$
Zusammenhang zwischen Druck-, Dichte- und Temperaturschwankungen: $\frac{{{p^\prime }}}{{\bar p}} = \frac{{{T^\prime }}}{{\bar T}} + \frac{{{\rho ^\prime }}}{{\bar \rho }}$
Morkovin-Hypothese: $\frac{{{p^\prime }}}{{\bar p}} \ll 1,\quad \frac{{T_0^\prime }}{{{{\bar T}_0}}} \ll 1\quad \to \quad \frac{{{\rho ^\prime }}}{{\bar \rho }} \approx - \frac{{{T^\prime }}}{{\bar T}} \approx \left( {\kappa - 1} \right)M{a^2}\frac{{{u^\prime }}}{{\bar u}}$
Grenzschichtannahmen: $\bar v \ll \bar u\quad ,\quad \frac{\partial }{{\partial x}} \ll \frac{\partial }{{\partial y}}$
Kontinuitätsgleichung für im Mittel stationäre und zweidimensionale turbulente kompressible Strömungen:

$\frac{\partial }{{\partial x}}\left( {\bar \rho \bar u} \right) + \frac{\partial }{{\partial y}}\left( {\overline {\rho v} } \right) = 0$
Turbulente Tangentialspannung für kompressible Strömungen: ${\tau _{tur}} = - \bar \rho \overline {{u^\prime }{v^\prime }}$
Gleichungssystem für ebene, im Mittel stationäre turbulente kompressible Grenzschichtströmungen:

$\begin{gathered} \frac{\partial }{{\partial x}}\left( {\bar \rho \bar u} \right) + \frac{\partial }{{\partial y}}\left( {\bar \rho \bar v} \right) = 0 \hfill \\ \bar \rho \bar u\frac{{\partial \bar u}}{{\partial x}} + \bar \rho \bar v\frac{{\partial \bar u}}{{\partial y}} = - \frac{{d{p_{^\delta }}}}{{dx}} + \frac{\partial }{{\partial y}}\left( {\mu \frac{{\partial \bar u}}{{\partial y}} - \bar \rho \overline {{u^\prime }{v^\prime }} } \right) \hfill \\ {c_p}\left( {\bar \rho \bar u\frac{{\partial \bar T}}{{\partial x}} + \overline {\rho v} \frac{{\partial \bar T}}{{\partial y}}} \right) = \bar u\frac{{d{p_{^\delta }}}}{{dx}} + \frac{\partial }{{\partial y}}\left( {\lambda \frac{{\partial \bar T}}{{\partial y}} - \bar \rho {c_p}\overline {{T^\prime }{v^\prime }} } \right) + \left( {\mu \frac{{\partial \bar u}}{{\partial y}} - \bar \rho \overline {{u^\prime }{v^\prime }} } \right)\frac{{\partial \bar u}}{{\partial y}} \hfill \\ \end{gathered}$

Randbedingungen:

$\begin{gathered} y = 0:\quad \bar u = \bar \rho \bar v = 0\quad ,\quad \bar T = {T_w} \hfill \\ y = \delta :\quad \bar u = {u_\delta }\left( x \right)\quad ,\quad \bar T = {T_\delta }\left( x \right)\quad ,\quad \bar \rho = {\rho _\delta }\left( x \right) \hfill \\ \end{gathered}$