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

    Amplitude / Intensität der Reflektierten Welle: {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} (VQ: 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\Rightarrowid. 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}

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