pttools.bubble.approx

Functions for calculating approximate solutions

Functions

pttools.bubble.approx.A2_approx(xi0)

Approximate solution for A2. \(A2_\text{approx} = \frac{3(2\xi_0 - 1)}{1 - \xi_0^2}\)

Parameters:: xi0 (float | float64 | ndarray) – \(\xi_0\)
Returns:: A2
Return type:: float | float64 | ndarray

pttools.bubble.approx.v_approx_high_alpha(xi, v_wall, v_xi_wall)

Approximate solution for fluid velocity \(v(\xi)\) near \(v(\xi) = \xi\).

Parameters:

xi (float | float64 | ndarray) – \(\xi\)
v_wall (float | float64 | ndarray) – \(v_\text{wall}\)
v_xi_wall (float | float64 | ndarray) – \(v(\xi_\text{wall})\)

Returns:

\(v_\text{approx}\)

Return type:

float | float64 | ndarray

pttools.bubble.approx.v_approx_hybrid(xi, v_wall, v_xi_wall)

Approximate solution for fluid velocity \(v(\xi)\) near \(v(\xi) = \xi\). Same as v_approx_high_alpha().

Parameters:

xi (float | float64 | ndarray) – \(\xi\)
v_wall (float | float64 | ndarray) – \(v_\text{wall}\)
v_xi_wall (float | float64 | ndarray) – \(v(\xi_\text{wall})\)

Returns:

\(v_\text{approx}\)

Return type:

float | float64 | ndarray

pttools.bubble.approx.v_approx_low_alpha(xi, v_wall, alpha)

Approximate solution for fluid velocity \(v(\xi)\) at low \(\alpha_+ = \alpha_n\).

Xi:

\(\xi\)

V_wall:

\(v_\text{wall}\)

Alpha:

\(\alpha\)

Returns:

\(v_\text{approx}\)

Parameters:

xi (ndarray)
v_wall (float)
alpha (float)

Return type:

ndarray

pttools.bubble.approx.w_approx_high_alpha(xi, v_wall, v_xi_wall, w_xi_wall)

Approximate solution for enthalpy \(w(\xi)\) near \(v(\xi) = \xi\).

Parameters:

xi (float | float64 | ndarray) – \(\xi\)
v_wall (float | float64 | ndarray) – \(v_\text{wall}\)
v_xi_wall (float | float64 | ndarray) – \(v(\xi_\text{wall})\)
w_xi_wall (float | float64 | ndarray) – \(w(\xi_\text{wall})\)

Returns:

\(w_\text{approx}\)

Return type:

float | float64 | ndarray

pttools.bubble.approx.w_approx_low_alpha(xi, v_wall, alpha)

Approximate solution for enthalpy \(w(\xi)\) at low \(\alpha_+ = \alpha_n\). (Not complete for \(\xi < \min(v _\text{wall}, cs_0)\)).

Parameters:

xi (ndarray) – \(\xi\)
v_wall (float) – \(v_\text{wall}\)
alpha (float) – \(\alpha\)

Returns:

\(w_\text{approx}\)

Return type:

ndarray

pttools.bubble.approx.xi_zero(v_wall, v_xi_wall)

Used in approximate solution near \(v(\xi) = \xi\): defined as solution to \(v(\xi_0) = \xi_0\).

\[\xi_0 = \frac{1}{3} (v(\xi_\text{wall}+2v_\text{wall})\]

Parameters:

v_wall (float | float64 | ndarray) – \(v_\text{wall}\)
v_xi_wall (float | float64 | ndarray) – \(v(\xi_\text{wall})\)

Returns:

\(\xi_0\)

Return type:

float | float64 | ndarray