3.1 Asymptotic equations of flow behaviour

Substituting (3.1)–(3.4) into (2.3)–(2.5), (2.7) and (2.9) results in the following equations relating the perturbation functions:

from (2.3)

(3.5)$$\begin{eqnarray}\unicode[STIX]{x1D716}^{2/3}(\bar{\unicode[STIX]{x1D70C}}_{1}+\bar{u}_{1})_{\bar{x}}+\unicode[STIX]{x1D716}^{4/3}[(\bar{\unicode[STIX]{x1D70C}}_{1}\bar{u}_{1}+\bar{\unicode[STIX]{x1D70C}}_{2}+\bar{u}_{2})_{\bar{x}}+\bar{v}_{1{\tilde{y}}}]+\cdots =0,\end{eqnarray}$$

from (2.4)

(3.6)$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D716}^{2/3}\left(\frac{a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}\bar{u}_{1}+\bar{p}_{1}\right)_{\bar{x}}+\unicode[STIX]{x1D716}^{2/3}\frac{a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}(\bar{\unicode[STIX]{x1D70C}}_{1}+\bar{u}_{1})_{\bar{x}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D716}^{4/3}\left[\left(\frac{a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}(\bar{\unicode[STIX]{x1D70C}}_{2}+2\bar{u}_{2}+\bar{u}_{1}^{2}+2\bar{\unicode[STIX]{x1D70C}}_{1}\bar{u}_{1})+\bar{p}_{2}\right)_{\bar{x}}+\frac{a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}\bar{v}_{1{\tilde{y}}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\cdots =0,\end{eqnarray}$$

from (2.5)

(3.7)$$\begin{eqnarray}\unicode[STIX]{x1D716}\left(\bar{p}_{1{\tilde{y}}}+\frac{a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}\bar{v}_{1\bar{x}}\right)+\cdots =0,\end{eqnarray}$$

from (2.7)

(3.8)$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D716}^{2/3}\left[\bar{p}_{1}-\left(1+\frac{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}\right)\bar{T}_{1}-\left(\frac{p_{\infty }+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{(1-b\unicode[STIX]{x1D70C}_{\infty })p_{\infty }}-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}\right)\bar{\unicode[STIX]{x1D70C}}_{1}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D716}^{4/3}\left[\bar{p}_{2}-\left(1+\frac{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}\right)\bar{T}_{2}-\left(\frac{p_{\infty }+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{(1-b\unicode[STIX]{x1D70C}_{\infty })p_{\infty }}-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}\right)\bar{\unicode[STIX]{x1D70C}}_{2}\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left(\frac{p_{\infty }+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{(1-b\unicode[STIX]{x1D70C}_{\infty })p_{\infty }}\right)\bar{\unicode[STIX]{x1D70C}}_{1}\bar{T}_{1}\nonumber\\ \displaystyle & & \displaystyle \quad -\left.\left(\frac{b\unicode[STIX]{x1D70C}_{\infty }(p_{\infty }+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2})}{(1-b\unicode[STIX]{x1D70C}_{\infty })^{2}p_{\infty }}-\frac{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}\right)\bar{\unicode[STIX]{x1D70C}}_{1}^{2}+\left(\frac{p_{\infty }+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{(1-b\unicode[STIX]{x1D70C}_{\infty })p_{\infty }}-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}\right)\bar{g}_{1}\right]+\cdots =0,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and from (2.9)

(3.9)$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D716}^{2/3}\left[\left(1+\frac{R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })}\right)\bar{T}_{1}+M_{\infty }^{2}\frac{a_{\infty }^{2}}{C_{vv}T_{\infty }}\bar{u}_{1}+\left(\frac{b\unicode[STIX]{x1D70C}_{\infty }R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })^{2}}-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }}{C_{vv}T_{\infty }}\right)\bar{\unicode[STIX]{x1D70C}}_{1}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D716}^{4/3}\left[\left(1+\frac{R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })}\right)\bar{T}_{2}+M_{\infty }^{2}\frac{a_{\infty }^{2}}{C_{vv}T_{\infty }}\left(\bar{u}_{2}+\frac{\bar{u}_{1}^{2}}{2}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\,\left(\frac{b\unicode[STIX]{x1D70C}_{\infty }R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })^{2}}-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }}{C_{vv}T_{\infty }}\right)\bar{\unicode[STIX]{x1D70C}}_{2}+\frac{(b\unicode[STIX]{x1D70C}_{\infty })^{2}R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })^{3}}\bar{\unicode[STIX]{x1D70C}}_{1}^{2}+\frac{b\unicode[STIX]{x1D70C}_{\infty }R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })^{2}}\bar{T}_{1}\bar{\unicode[STIX]{x1D70C}}_{1}\nonumber\\ \displaystyle & & \displaystyle \quad -\left.\bar{g}_{1}\left(\frac{h_{fg}(T_{\infty })}{C_{vv}T_{\infty }}+\frac{b\unicode[STIX]{x1D70C}_{\infty }R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })^{2}}-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }}{C_{vv}T_{\infty }}\right)\right]+\cdots =0.\end{eqnarray}$$

Note that in (3.5) and (3.6), the term $\unicode[STIX]{x1D716}^{2/3}(\bar{\unicode[STIX]{x1D70C}}_{1}+\bar{u}_{1})$ may be $O(\unicode[STIX]{x1D716}^{4/3})$. The leading-order $O(\unicode[STIX]{x1D716}^{2/3})$ terms of (3.6) then give $\bar{p}_{1}+a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }\bar{u}_{1}/p_{\infty }=f({\tilde{y}})$. Owing to the uniformity of the upstream flow, $f({\tilde{y}})=0$. Therefore,

(3.10)$$\begin{eqnarray}\bar{p}_{1}=-\frac{a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}\bar{u}_{1}.\end{eqnarray}$$

The isentropic frozen speed of sound ($a_{\infty }$) in a van der Waals fluid can be expressed as (see Rusak & Wang Reference Rusak and Wang1997)

(3.11)$$\begin{eqnarray}a_{\infty }^{2}=\frac{p_{\infty }+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{(1-b\unicode[STIX]{x1D70C}_{\infty })\unicode[STIX]{x1D70C}_{\infty }}\left(1+\frac{R_{v}}{C_{vv}}\right)-2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }\end{eqnarray}$$

or

(3.12)$$\begin{eqnarray}\displaystyle \frac{a_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}=\frac{p_{\infty }+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{(1-b\unicode[STIX]{x1D70C}_{\infty })p_{\infty }}\left(1+\frac{R_{v}}{C_{vv}}\right)-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}. & & \displaystyle\end{eqnarray}$$

For the following analysis, the formula

(3.13)$$\begin{eqnarray}\displaystyle \frac{a_{\infty }^{2}}{C_{vv}T_{\infty }}=\frac{R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })^{2}}\left(1+\frac{R_{v}}{C_{vv}}\right)-\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }}{C_{vv}T_{\infty }} & & \displaystyle\end{eqnarray}$$

is also needed. To simplify the above equations, the following non-dimensional parameters are defined:

(3.14a-d)$$\begin{eqnarray}K_{r}=\frac{R_{v}}{C_{vv}(1-b\unicode[STIX]{x1D70C}_{\infty })},\quad K_{b}=\frac{b\unicode[STIX]{x1D70C}_{\infty }}{1-b\unicode[STIX]{x1D70C}_{\infty }},\quad K_{a}=\frac{2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }}{C_{vv}T_{\infty }},\quad K_{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{\infty }^{2}}{p_{\infty }}.\end{eqnarray}$$

Expressing the isentropic speed of sound in terms of these parameters:

(3.15a,b)$$\begin{eqnarray}\frac{a_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}=(1+K_{\unicode[STIX]{x1D6FC}})(1+K_{b}+K_{r})-2K_{\unicode[STIX]{x1D6FC}};\quad \frac{a_{\infty }^{2}}{C_{vv}T_{\infty }}=K_{r}(1+K_{b}+K_{r})-K_{a}.\end{eqnarray}$$

The leading-order $O(\unicode[STIX]{x1D716}^{2/3})$ terms in (3.8) and (3.9) together with the result of (3.10) give

(3.16)$$\begin{eqnarray}\bar{T}_{1}=-K_{r}M_{\infty }^{2}\bar{u}_{1}.\end{eqnarray}$$

Utilizing (3.10) and (3.16), the leading-order $O(\unicode[STIX]{x1D716}^{2/3})$ terms in (3.8) give

(3.17)$$\begin{eqnarray}\bar{\unicode[STIX]{x1D70C}}_{1}=-M_{\infty }^{2}\bar{u}_{1}.\end{eqnarray}$$

As a result, $\unicode[STIX]{x1D716}^{2/3}(\bar{\unicode[STIX]{x1D70C}}_{1}+\bar{u}_{1})=\unicode[STIX]{x1D716}^{2/3}(1-M_{\infty }^{2})\bar{u}_{1}=\unicode[STIX]{x1D716}^{4/3}K\bar{u}_{1}$ which validates the previous assumption. Therefore, in the higher order $O(\unicode[STIX]{x1D716}^{4/3})$, (3.5) becomes

(3.18)$$\begin{eqnarray}(\bar{\unicode[STIX]{x1D70C}}_{2}+\bar{u}_{2}+K\bar{u}_{1}-M_{\infty }^{2}\bar{u}_{1}^{2})_{\bar{x}}+\bar{v}_{1{\tilde{y}}}=0,\end{eqnarray}$$

and (3.6) becomes

(3.19)$$\begin{eqnarray}\left(\bar{p}_{2}+\frac{a_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}M_{\infty }^{2}[\bar{\unicode[STIX]{x1D70C}}_{2}+2\bar{u}_{2}+\bar{u}_{1}^{2}(1-2M_{\infty }^{2})+K\bar{u}_{1}]\right)_{\bar{x}}+\frac{a_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}M_{\infty }^{2}\bar{v}_{1{\tilde{y}}}=0.\end{eqnarray}$$

From (3.18) and (3.19)

(3.20)$$\begin{eqnarray}\displaystyle \bar{p}_{2}=-\frac{a_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }}{p_{\infty }}M_{\infty }^{2}[\bar{u}_{2}+\bar{u}_{1}^{2}(1-M_{\infty }^{2})], & & \displaystyle\end{eqnarray}$$

or

(3.21)$$\begin{eqnarray}\bar{p}_{2}=-[(1+K_{\unicode[STIX]{x1D6FC}})(1+K_{b}+K_{r})-2K_{\unicode[STIX]{x1D6FC}}]M_{\infty }^{2}[\bar{u}_{2}+\bar{u}_{1}^{2}(1-M_{\infty }^{2})].\end{eqnarray}$$

Higher-order $O(\unicode[STIX]{x1D716}^{4/3})$ terms in (3.9) give

(3.22)$$\begin{eqnarray}\displaystyle & & \displaystyle [K_{r}(1+K_{b}+K_{r})-K_{a}]M_{\infty }^{2}\left(\bar{u}_{2}+\frac{\bar{u}_{1}^{2}}{2}\right)+(1+K_{r})\bar{T}_{2}+(K_{b}K_{r}-K_{a})\bar{\unicode[STIX]{x1D70C}}_{2}+K_{r}K_{b}^{2}\bar{\unicode[STIX]{x1D70C}}_{1}^{2}\nonumber\\ \displaystyle & & \displaystyle \quad +\,K_{r}K_{b}\bar{\unicode[STIX]{x1D70C}}_{1}\bar{T}_{1}-\left(\frac{h_{fg}(T_{\infty })}{C_{vv}T_{\infty }}+K_{r}K_{b}-K_{a}\right)\bar{g}_{1}=0.\end{eqnarray}$$

Utilizing (3.16) and (3.17) in the above equation gives

(3.23)$$\begin{eqnarray}\displaystyle & & \displaystyle [K_{r}(1+K_{b}+K_{r})-K_{a}]M_{\infty }^{2}\frac{\bar{u}_{1}^{2}}{2}+(1+K_{r})(\bar{T}_{2}+K_{r}M_{\infty }^{2}\bar{u}_{2})+(K_{b}K_{r}-K_{a})(\bar{\unicode[STIX]{x1D70C}}_{2}+M_{\infty }^{2}\bar{u}_{2})\nonumber\\ \displaystyle & & \displaystyle \quad +\,K_{r}K_{b}(K_{r}+K_{b})M_{\infty }^{4}\bar{u}_{1}^{2}-\left(\frac{h_{fg}(T_{\infty })}{C_{vv}T_{\infty }}+K_{r}K_{b}-K_{a}\right)\bar{g}_{1}=0.\end{eqnarray}$$

Higher-order $O(\unicode[STIX]{x1D716}^{4/3})$ terms in (3.8) give

(3.24)$$\begin{eqnarray}\displaystyle & & \displaystyle \bar{p}_{2}-(1+K_{\unicode[STIX]{x1D6FC}})\bar{T}_{2}-[(1+K_{\unicode[STIX]{x1D6FC}})(1+K_{b})-2K_{\unicode[STIX]{x1D6FC}}]\bar{\unicode[STIX]{x1D70C}}_{2}- [(1+K_{\unicode[STIX]{x1D6FC}})(1+K_{b})(K_{b}+K_{r})\nonumber\\ \displaystyle & & \displaystyle \quad -\,K_{\unicode[STIX]{x1D6FC}}\!]M_{\infty }^{4}\bar{u}_{1}^{2}+[(1+K_{\unicode[STIX]{x1D6FC}})(1+K_{b})-2K_{\unicode[STIX]{x1D6FC}}]\bar{g}_{1}=0.\end{eqnarray}$$

Equations (3.18) and (3.21)–(3.24) together result in

(3.25)$$\begin{eqnarray}(K-K_{G}M_{\infty }^{2}\bar{u}_{1})\bar{u}_{1\bar{x}}+\bar{v}_{1{\tilde{y}}}=\bar{g}_{1\bar{x}}\left(\frac{h_{fg}(T_{\infty })}{K_{3}C_{vv}T_{\infty }(1+K_{r})}-1\right).\end{eqnarray}$$

Here,

(3.26)$$\begin{eqnarray}\displaystyle K_{G}=2\left(1+\frac{K_{1}+K_{2}}{K_{3}}\right), & & \displaystyle\end{eqnarray}$$

where

(3.27)$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle K_{1}=-\frac{K_{r}}{2}-\frac{K_{b}K_{r}-K_{a}}{2(1+K_{r})},\quad K_{2}=-\frac{K_{\unicode[STIX]{x1D6FC}}}{1+K_{\unicode[STIX]{x1D6FC}}}+(1+K_{b})(K_{b}+K_{r})-K_{b}K_{r}\frac{K_{b}+K_{r}}{1+K_{r}},\\ \displaystyle K_{3}=(1+K_{b})-\frac{K_{b}K_{r}-K_{a}}{1+K_{r}}-2\frac{K_{\unicode[STIX]{x1D6FC}}}{1+K_{\unicode[STIX]{x1D6FC}}}.\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Also, from (3.7) and (3.10), it is found that

(3.28)$$\begin{eqnarray}\bar{u}_{1{\tilde{y}}}-\bar{v}_{1\bar{x}}=0.\end{eqnarray}$$

The set of equations (3.25) and (3.28) constitute an extended Kármán–Guderley system for transonic small-disturbance flow of steam with condensation. Equation (3.25) contains a heat source term which depends on the condensate mass fraction perturbation function, $\bar{g}_{1}$. It can be concluded from (3.28) that the flow has no vorticity and, hence, is isentropic to the leading order $O(\unicode[STIX]{x1D716}^{2/3})$ of perturbations in temperature, density, pressure and velocity. This allows for the definition of a velocity perturbation potential function $\unicode[STIX]{x1D719}_{1}$ where $\bar{u}_{1}=\unicode[STIX]{x1D719}_{1\bar{x}}$ and $\bar{v}_{1}=\unicode[STIX]{x1D719}_{1{\tilde{y}}}$. Then, (3.25) can be written as

(3.29)$$\begin{eqnarray}(K-K_{G}M_{\infty }^{2}\unicode[STIX]{x1D719}_{1\bar{x}})\unicode[STIX]{x1D719}_{1\bar{x}\bar{x}}+\unicode[STIX]{x1D719}_{1{\tilde{y}}{\tilde{y}}}=\bar{g}_{1\bar{x}}\left(\frac{h_{fg}(T_{\infty })}{K_{3}C_{vv}T_{\infty }(1+K_{r})}-1\right).\end{eqnarray}$$

Equation (3.29) is also referred to as the extended TSD equation.

Substitution of the asymptotic expansions (3.1) in airfoil boundary conditions (2.14), the far-field conditions (2.15) and the Kutta condition provide the boundary conditions for the extended TSD equation

(3.30)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D719}_{1{\tilde{y}}}(\bar{x},0^{+})=F_{u}^{\prime }(\bar{x})\quad \text{and}\quad \unicode[STIX]{x1D719}_{1{\tilde{y}}}(\bar{x},0^{-})=F_{l}^{\prime }(\bar{x})\quad \text{for }0\leqslant \bar{x}\leqslant 1,\\ \displaystyle \unicode[STIX]{x1D719}_{1\bar{x}},\unicode[STIX]{x1D719}_{1{\tilde{y}}}\rightarrow 0\quad \text{as }\bar{x}\rightarrow -\infty ,\quad \unicode[STIX]{x1D719}_{1\bar{x}}(1,0^{-})=\unicode[STIX]{x1D719}_{1\bar{x}}(1,0^{+}).\end{array}\right\}\end{eqnarray}$$

At low pressures and temperatures, the water vapour thermodynamic behaviour approaches the perfect-gas behaviour. Under these limiting conditions, $\unicode[STIX]{x1D6FC}=b=0$, and therefore, $K_{\unicode[STIX]{x1D6FC}}=K_{a}=K_{b}=0$, $K_{G}=\unicode[STIX]{x1D6FE}_{v}+1$, $K_{3}=1$ and $1+K_{r}=\unicode[STIX]{x1D6FE}_{v}$, where $\unicode[STIX]{x1D6FE}_{v}$ is the specific heat ratio of water vapour. Equation (3.29) then becomes

(3.31)$$\begin{eqnarray}(K-(\unicode[STIX]{x1D6FE}_{v}+1)M_{\infty }^{2}\unicode[STIX]{x1D719}_{1\bar{x}})\unicode[STIX]{x1D719}_{1\bar{x}\bar{x}}+\unicode[STIX]{x1D719}_{1{\tilde{y}}{\tilde{y}}}=\bar{g}_{1\bar{x}}\left(\frac{h_{fg}(T_{\infty })}{C_{pv}T_{\infty }}-1\right),\end{eqnarray}$$

which is the TSD equation for perfect-gas steam flow with non-equilibrium and homogeneous condensation (see Virk & Rusak Reference Virk and Rusak2019).

3.2 Asymptotic equations of the condensation process

Substituting the asymptotic expansions (3.1) into the set of condensation equations (2.10)–(2.13) gives the relationships between the leading-order $O(\unicode[STIX]{x1D716}^{4/3})$ perturbation functions of the condensation parameters, $\bar{g}_{1},\bar{Q}_{11},\bar{Q}_{21}$ and $\bar{Q}_{31}$,

(3.32)$$\begin{eqnarray}\displaystyle & \displaystyle \bar{g}_{1\bar{x}}=4\unicode[STIX]{x03C0}K_{t}\left(\bar{Q}_{11}\frac{\text{d}\bar{r}}{\text{d}\bar{t}}+\frac{1}{3}\bar{J}_{1}\bar{r}_{0}^{3}\right), & \displaystyle\end{eqnarray}$$

(3.33)$$\begin{eqnarray}\displaystyle & \displaystyle \bar{Q}_{11\bar{x}}=K_{t}\left(2\bar{Q}_{21}\frac{\text{d}\bar{r}}{\text{d}\bar{t}}+\bar{J}_{1}\bar{r}_{0}^{2}\right), & \displaystyle\end{eqnarray}$$

(3.34)$$\begin{eqnarray}\displaystyle & \displaystyle \bar{Q}_{21\bar{x}}=K_{t}\left(\bar{Q}_{31}\frac{\text{d}\bar{r}}{\text{d}\bar{t}}+\bar{J}_{1}\bar{r}_{0}\right), & \displaystyle\end{eqnarray}$$

(3.35)$$\begin{eqnarray}\displaystyle & \displaystyle \bar{Q}_{31\bar{x}}=K_{t}\bar{J}_{1}. & \displaystyle\end{eqnarray}$$

Equations (3.32)–(3.35) form a complex set of first-order and closed-coupled nonlinear ordinary differential equations. In these equations, $\bar{r}_{0}=r_{0}/l_{c}$ while $\text{d}\bar{r}/\text{d}\bar{t}$ and $\bar{J}_{1}$ are non-dimensional forms of $\text{d}r/\text{d}t$ and $J$ (see appendix for expressions). It should be noted that $\bar{J}_{1}$ depends on the number of water molecules in a characteristic droplet $n_{c}=(4\unicode[STIX]{x03C0}/3)\unicode[STIX]{x1D70C}_{f\infty }l_{c}^{3}/m$. Here, $m$ is the mass of a water molecule. Note that $n_{c}$ is a function of upstream temperature ($T_{\infty }$) only. Also, the nucleation rate $\bar{J}_{1}$ is a highly sensitive function of $\bar{T}$, that is given by

(3.36)$$\begin{eqnarray}\bar{T}=1-\unicode[STIX]{x1D716}^{2/3}K_{r}M_{\infty }^{2}\unicode[STIX]{x1D719}_{1\bar{x}}+\unicode[STIX]{x1D716}^{4/3}\left(1+K_{b}-\frac{2K_{\unicode[STIX]{x1D6FC}}}{1+K_{\unicode[STIX]{x1D6FC}}}\right)\left(\frac{h_{fg}(T_{\infty })}{K_{3}C_{vv}T_{\infty }}\right)\bar{g}_{1}+\cdots \,.\end{eqnarray}$$

Note that $\bar{T}$ depends on $\unicode[STIX]{x1D719}_{1\bar{x}}$ and $\bar{g}_{1}$; it is necessary for accuracy of the computations to account for the effects of changes in $\bar{g}_{1}$ (of $O(\unicode[STIX]{x1D716}^{4/3})$) when computing the temperature field. In addition, in (3.32)–(3.35), $K_{t}$ is the condensation similarity parameter,

(3.37)$$\begin{eqnarray}K_{t}=\frac{t_{f}}{t_{c}}=\frac{cp_{\infty }}{\unicode[STIX]{x1D70E}(T_{\infty })M_{\infty }a_{\infty }}\sqrt{\frac{R_{v}T_{\infty }}{8\unicode[STIX]{x03C0}}},\end{eqnarray}$$

where $t_{f}=c/U_{\infty }$ represents the characteristic convection time and $t_{c}=l_{c}/u_{c}$ represents the characteristic condensation time. If $K_{t}<1$, time spent by the steam to convect across the airfoil surface is less than the time spent by the steam to condense, and thus no condensation is observed. Conversely, if $K_{t}$ is very large (i.e. time required for initiation of condensation is much less than the time required for the flow to convect over the airfoil surface) then condensation might be observed along the airfoil surface.

The far-field conditions for the condensation equations

(3.38)$$\begin{eqnarray}\bar{g}_{1}=\bar{Q}_{11}=\bar{Q}_{21}=\bar{Q}_{31}=0,\quad \text{as }\bar{x}\rightarrow -\infty ,\end{eqnarray}$$

provide the initial values for integration of (3.32)–(3.35).

3.3 Summary of asymptotic model

In summary, the theoretical transonic small-disturbance model describes the flow and condensation fields of pure water vapour around a thin airfoil with a nonlinear partial differential equation (3.29) coupled with a set of ordinary differential equations (3.32)–(3.35). The numerical solution of these equations requires a coupled iterative process which provides the fields of $\unicode[STIX]{x1D719}_{1}(\bar{x},{\tilde{y}})$ and of $\bar{g}_{1}(\bar{x},{\tilde{y}})$. The field of $\unicode[STIX]{x1D719}_{1}$ results in the pressure field, $\bar{p}=p/p_{\infty }=1-\unicode[STIX]{x1D716}^{2/3}(a_{\infty }^{2}M_{\infty }^{2}\unicode[STIX]{x1D70C}_{\infty }/p_{\infty })\unicode[STIX]{x1D719}_{1\bar{x}}+\cdots \,$, the axial velocity field $u/U_{\infty }=1+\unicode[STIX]{x1D716}^{2/3}\unicode[STIX]{x1D719}_{1\bar{x}}+\cdots \,$, the transverse velocity field $v/U_{\infty }=\unicode[STIX]{x1D716}\unicode[STIX]{x1D719}_{1\bar{y}}+\cdots \,$ and, together with $\bar{g}_{1}$, the temperature field according to (3.36). The field of condensate mass fraction ($g$) is obtained as $g=\unicode[STIX]{x1D716}^{4/3}\bar{g}_{1}+\cdots \,$. The pressure coefficient ($C_{p}$) is computed as $C_{p}=(p-p_{\infty })/(\frac{1}{2}\unicode[STIX]{x1D70C}_{\infty }U_{\infty }^{2})=-2\unicode[STIX]{x1D716}^{2/3}\unicode[STIX]{x1D719}_{1\bar{x}}+\cdots \,$. The wave drag and lift coefficients, $C_{d}$ and $C_{l}$, can be obtained by the integration of pressure distributions along the airfoil surfaces as

(3.39a,b)$$\begin{eqnarray}C_{d}=\unicode[STIX]{x1D716}\int _{0}^{1}[(C_{p})_{u}F_{u}^{\prime }-(C_{p})_{l}F_{l}^{\prime }]\,\text{d}\bar{x},\quad C_{l}=\int _{0}^{1}[(C_{p})_{l}-(C_{p})_{u}]\,\text{d}\bar{x},\end{eqnarray}$$

where $u,l$ represent values on the upper and lower surfaces of airfoil. The model also identifies the similarity parameters of the flow problem which are as follows: thickness ratio of airfoil $\unicode[STIX]{x1D716}$, scaled angle of attack $\unicode[STIX]{x1D6E9}$, classical transonic similarity parameter $K$, real-gas similarity parameter $K_{G}$, scaled latent heat of condensation $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]$, upstream flow supersaturation ratio $S_{\infty }$, ratio of time scales of convection to condensation $K_{t}$ and number of molecules in a characteristic droplet $n_{c}$.

We note that the extended TSD equation (3.29) displays a leading-edge singularity as the flow approaches stagnation near the leading edge ($\bar{x}\rightarrow 0,{\tilde{y}}\rightarrow 0$). The singularity affects the TSD solution in a small region $O(\unicode[STIX]{x1D716}^{2})$ around the airfoil’s nose. As flow becomes transonic and the upstream flow Mach number approaches unity, the nose singularity causes the flow to be nearly symmetric about the leading edge and with respect to subsonic flows, induces a loss of leading-edge suction and lift. In regions away from the leading edge of the order of $\unicode[STIX]{x1D716}$ and lower, the TSD model solution is not affected by the nose singularity. In the present study, the nose singularity is removed from the TSD solution with the multiscale matched-asymptotic approach of Rusak (Reference Rusak1993) around the leading edge by replacing $\unicode[STIX]{x1D6FE}_{v}+1$ with $K_{G}$ in that analysis. After removal of the nose singularity, the TSD model solution was found to match better with an Euler flow problem solution around the nose region. Note, however, that in the nose region the temperature is high, and no condensation occurs.

5.1 Grid convergence studies

Convergence of the current numerical algorithm upon mesh refinement is studied first. We focus on a representative example of a steady wet steam flow around a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) at zero angle of attack that is computed using several progressively refined meshes. The uniform upstream conditions of this flow problem are described by $T_{\infty }=450~\text{K}$, $p_{\infty }=1.2~\text{MPa}$ ($S_{\infty }=1.28$) and $M_{\infty }=0.8$. Stagnation conditions of the upstream fluid are given by $T_{0}=501~\text{K}$, $p_{0}=1.87~\text{MPa}$ and $\unicode[STIX]{x1D6F7}_{0}=69.4\,\%$. For this case, the similarity parameters are $K=1.48,h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]=2.035,K_{G}=2.20,n_{c}=18.14$ and $K_{t}=5.88\times 10^{5}$. For this problem, the flow and thermodynamic fields are symmetrical about the $x$ axis, so only the upper half of the computational flow domain is considered with dimensions $3c$ (along the $\bar{x}$ axis) and $3c$ (along the ${\tilde{y}}$ axis). The upstream boundary of the domain is placed $1c$ ahead of the leading edge of the airfoil; the downstream boundary is placed $1c$ behind the trailing edge; the top boundary is placed $3c$ over the airfoil whereas the bottom boundary coincides with the chord of the airfoil. The different mesh sizes chosen for the convergence study are: 300 (along the $\bar{x}$ axis) $\times$ 150 (along the ${\tilde{y}}$ axis), $300\times 200$, $450\times 200$, $450\times 300$ and $600\times 300$. The computed distributions of the pressure coefficient ($-C_{p}$) and the condensate mass fraction ($g$) at the airfoil surface for the selected meshes are presented in figures 3 and 4, respectively. Figure 5 shows the pressure–temperature ($p{-}T$) phase diagram for a streamline along the airfoil surface for several mesh sizes. Also shown for reference in figure 5 is the vapour–liquid saturation pressure ($p_{g}$) line. The numerical results converge upon mesh refinement. From figures 3, 4 and 5, it can be noticed that the numerical solutions of $-C_{p},g$ and the $p{-}T$ line computed by the various meshes are close to each other. The numerical results from using a $450\times 300$ mesh are within 1 % of the corresponding numerical results from using a finer mesh i.e. $600\times 300$. It is therefore concluded that a 450 $\times$ 300 mesh provides grid-converged computations for non-lifting wet steam flows around airfoils. Figures 6 and 7 show the distributions of local supersaturation ratio ($S$) and nucleation rate ($J$) along a streamline close to the airfoil surface for the chosen mesh, respectively.

Figure 3. Distribution of $-C_{p}$ at the NACA0012 airfoil surface for wet steam flow at $S_{\infty }=1.28$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$ using several meshes (points 1–3 and 9 are beyond the scale of the figure).

Figure 4. Distribution of $g$ at the NACA0012 airfoil surface for wet steam flow at $S_{\infty }=1.28$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$ using several meshes (points 1–3 are beyond the scale of the figure).

Figure 5. The $p{-}T$ phase diagram along the NACA0012 airfoil surface for wet steam flow at $S_{\infty }=1.28$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$ using several meshes.

Enlarging the computational domain along the $\bar{x}$ and ${\tilde{y}}$ directions provides numerical results which are within 2 % of the current numerical results. Using a far-field analytical approximation of the solution (Krupp & Murman Reference Krupp and Murman1972) on the boundaries of the computational domain for non-lifting flows, also gives numerical results within 2 % of the current TSD results. Similar behaviour was observed in past numerical studies of dry air flow with TSD theory (Cole & Cook Reference Cole and Cook1986). For flow problems with $K<1.5$ and $\unicode[STIX]{x1D6E9}<0.3$, the supersonic regions of dry air flows over NACA0012 airfoils lie below $\bar{y}=1$ (Krupp Reference Krupp1972). The computational domain with dimensions $3$ (along $\bar{x}$ axis) and $3$ (along ${\tilde{y}}$ axis) provides a good balance between accuracy of predictions and computational costs.

Figure 6. Distribution of supersaturation ratio ($S$) along a streamline close to the NACA0012 airfoil surface for wet steam flow at $S_{\infty }=1.28$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$ (point 1 is beyond the scale of the figure).

Figure 7. Distribution of nucleation rate ($J$) along a streamline close to the NACA0012 airfoil surface for wet steam flow at $S_{\infty }=1.28$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$ (points 1–3 and 7–9 are beyond the scale of the figure).

Figures 3–7 help in understanding the physical and thermodynamic behaviour of the flow. Important characteristic states in the flow are marked with points numbered from 1 to 9. The stagnation reservoir conditions are denoted by point 1 ($T_{0}=501~\text{K}$, $p_{0}=1.87~\text{MPa}$ and $\unicode[STIX]{x1D6F7}_{0}=69.4\,\%$). The upstream flow (point 2) ahead of the airfoil ($\bar{x}<0$) is decelerated, compressed and reaches stagnation at the airfoil’s leading edge (point 3). This results in maximum values of local pressure and temperature at the leading edge. Flow is in a superheated state in this region, preventing condensation occurring (see point 3 in figure 6). Flow then expands as it accelerates along the airfoil surface. As velocity increases, local pressure and temperature decrease, which leads to the flow becoming supersaturated ($S>1$). Substantial homogeneous nucleation is initiated on the airfoil surface (see point 4 in figure 7). Flow continues to accelerate and becomes sonic (point 5). Flow velocity increases and pressure decreases until it reaches minimum values of local pressure and temperature ($p=0.83~\text{MPa}$, $T=424~\text{K}$) at $\bar{x}\sim 0.15$ (point 6). The flow attains the maximum supersaturation ratio (see point 6 in figure 6) at this point, which is commonly referred to as the Wilson point. This marks the onset of macroscopic heat release due to condensation and related compression of the flow.

Condensate mass fraction $g$ increases from 0 in a nearly constant pressure process. The temperature is increased from 424 K at $\bar{x}=0.15$ to 445 K at $\bar{x}=0.4$ (point 7) at which the flow is nearly saturated ($p\sim p_{g}(T)$). Condensate mass fraction also attains its maximum value ($g\sim 0.027$). At $\bar{x}\sim 0.4$ (point 7), a shock wave occurs and the flow is compressed from $p=0.83~\text{MPa}$, $T=445~\text{K}$ to $p=1.04~\text{MPa}$, $T=459~\text{K}$ (point 8) and liquid droplets evaporate across the shock wave. The flow exhibits the classical Zierep shock wave behaviour downstream of the shock wave for $0.41\leqslant \bar{x}\leqslant 0.54$ where the pressure and temperature decrease and then increase with distance from the shock wave. Yet, evaporation continues and $g$ decreases in this range. For $0.54<\bar{x}<1$, water vapour compresses at near saturation conditions and evaporation of liquid droplets continues as flow decelerates along the airfoil towards the trailing edge (point 9). Behind the trailing edge, flow is accelerated, pressure and temperature decrease and condensation is re-initiated because $S$ becomes greater than 1 (see point 9 in figure 6). Far downstream of the airfoil, $g$ reaches a nearly constant value of 0.013 and a wake of saturated liquid trails behind the airfoil. All other flow cases presented in the following sections and figures follow similar flow and condensation processes.

Figure 8. Contours of $C_{p}$ (solid line) and $g$ (dashed line) for wet steam flow around a NACA0012 airfoil ($c=0.1~\text{m}$) at $S_{\infty }=1.28$, $M_{\infty }=0.8$, $T_{\infty }=450~\text{K}$ and $\unicode[STIX]{x1D6E9}=0$.

Figure 8 shows the mesh-converged solution of the contours of the pressure coefficient $C_{p}$ (solid contours) and the condensate mass fraction $g$ (dashed contours) in the airfoil’s near field at the given flow conditions. The flow around the shock wave at $x\sim 0.4c$ is apparent from the coalescence of $C_{p}$ contour lines. The shock wave also spreads in the transverse direction up to $y\sim 0.03~\text{m}$. The condensation region starts on the airfoil surface at $x\sim 0.15c$ and spreads downstream as well as in the transverse direction to form a saturated parallel condensation zone with transverse size $y\sim 0.03~\text{m}$.

For flows around airfoils at small angles of attack, a rectangular computational domain centred around the airfoil with dimensions $3c$ (along $\bar{x}$ axis) and $6c$ (along ${\tilde{y}}$ axis) is used. The upstream boundary of this domain is placed $1c$ ahead of the leading edge of the airfoil; the downstream boundary is placed $1c$ behind the trailing edge; the top boundary is placed $3c$ above the airfoil whereas the bottom boundary is placed $3c$ below the airfoil. Grid convergence studies for a flow problem with upstream conditions as described above and with $\unicode[STIX]{x1D703}=1.5^{\circ }$ reveal that a $450\times 600$ mesh provides grid-converged results and this mesh is used in all computations of lifting wet steam flows around airfoils with $|\unicode[STIX]{x1D703}|\leqslant 2^{\circ }$.

We also emphasize that, when compared with homogeneously condensing flow of humid air around the same airfoils (such as circular arcs) in the same Mach number regime ($M_{\infty }\sim 0.8$) and operating at similar flow conditions, diabatic compressions inside local supersonic regimes in steam flow are much weaker. For example, the present computations with external steam flows around airfoils do not depict any local recompression regions or two separated local supersonic regimes terminated by shock waves, as was found in the numerical computations of Schnerr & Dohrmann (Reference Schnerr and Dohrmann1990, Reference Schnerr and Dohrmann1994). The steam flow case is different from the humid air flow case. The difference is due to several reasons. First, since in the steam flow case there is no carrier gas of the water content, the right-hand side of the TSD equation is different from that of the humid air case; compare (3.29) of the present paper with (35) in Rusak & Lee (Reference Rusak and Lee2000a). The heat release of condensation goes from steam to air in the humid air case while in the steam flow case, it goes directly from the condensate to the water vapour, which has a lower specific heat ratio $\unicode[STIX]{x1D6FE}_{v}$ than air. Second, all the humid air cases are characterized by upstream pressure that is atmospheric and below and temperatures are relatively low (below 270 K). However, in the steam flow cases studied in this paper, upstream pressure and temperature are much higher and therefore characterized by higher $C_{vv}$, lower $\unicode[STIX]{x1D6FE}_{v}$ and lower $h_{fg}$ which lead to much weaker nonlinear effects and heat release from condensation to the vapour.

5.2 Computed examples

In engineering applications that employ high-speed steam flows, it is necessary to predict the flow behaviour and the resulting wave drag of the airfoils or blades, in an attempt to reduce the drag for higher energy efficiency of the system. The upstream flow and thermodynamic conditions of steam and the airfoil’s geometry and angle of attack are the primary parameters that determine the real-gas flow and condensation fields, and consequently the airfoil’s wave drag. With this purpose, the present TSD model of steam flow is used to conduct parametric studies where each one of these parameters is varied independently and their effects on flow and condensation physics are analysed.

First, the effects of varying upstream temperature ($T_{\infty }$) in the range of 375–450 K at fixed values of $S_{\infty }=1.3$ and $M_{\infty }=0.8$ for wet steam flow over a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) at $\unicode[STIX]{x1D6E9}=0$ are studied. Table 1 lists the values of similarity parameters $n_{c}$, $K_{t}$, $K_{G}$, $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]$, upstream pressure ($p_{\infty }$) and the stagnation properties of the upstream fluid ($p_{0}$, $T_{0}$ and $\unicode[STIX]{x1D6F7}_{0}$) for several flow cases. Note that $K=1.48$ for all the cases. From table 1, it can be noticed that $K_{G}$ decreases slightly as a result of a change in $T_{\infty }$ within the given range, whereas the changes in $n_{c}$, $K_{t}$, $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]$ and $p_{\infty }$ are more pronounced. It is therefore expected that condensation increases as $T_{\infty }$ increases at the described conditions.

Figure 9. Distribution of $-C_{p}$ at the NACA0012 airfoil surface for wet steam flow at $S_{\infty }=1.3$, $M_{\infty }=0.8$, $\unicode[STIX]{x1D6E9}=0$ and several $T_{\infty }$.

Table 1. Values of $n_{c},K_{t},K_{G},h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]$, $p_{\infty }$, $p_{0}$, $T_{0}$ and $\unicode[STIX]{x1D6F7}_{0}$ for various $T_{\infty }$ at the given conditions.

Figure 10. Distribution of $g$ at the NACA0012 airfoil surface for wet steam flow at $S_{\infty }=1.3$, $M_{\infty }=0.8$, $\unicode[STIX]{x1D6E9}=0$ and several $T_{\infty }$.

Figures 9 and 10 depict the $-C_{p}$ and $g$ distributions at the airfoil surface. A $p{-}T$ phase diagram for a streamline along the airfoil surface for the various flow cases is presented in figure 11. Also shown for reference in figure 11 is the vapour–liquid saturation pressure ($p_{g}$) line to help visualize the regions where the flow is supersaturated. From figures 9 and 10, it can be observed that the condensation region on the airfoil surface increases in terms of axial extent and magnitude of $g$ with an increase in $T_{\infty }$. At $T_{\infty }=375~\text{K}$, the condensation region and the amount of condensate mass fraction are relatively small and affect the pressure and temperature distributions negligibly. At $T_{\infty }=450~\text{K}$, the region of condensation and size of $g$ are significant and affect the pressure and temperature distributions notably. As $T_{\infty }$ increases, the characteristic length of condensation $l_{c}$ decreases (see (3.2)) which causes $n_{c}$ to decrease. This leads to an increase in nucleation rate $\bar{J}_{1}$. Also, $u_{c}$ increases (see (3.2)); this, along with the decrease in $l_{c}$, reduces the characteristic condensation time $t_{c}$. This causes a considerable increase in $K_{t}$ (see (3.37)). Combined effects of changes in $\bar{J}_{1}$ and $K_{t}$ result in increased condensation as $T_{\infty }$ increases. It can also be observed that the point of condensation onset moves upstream along the airfoil surface with an increase in $T_{\infty }$. The effect of condensation on the pressure field is clearly visible in figure 9. With increase in $T_{\infty }$, as the condensation region over the airfoil increases, the flow experiences increased compression in the supersonic region as a consequence of heat addition due to condensation of water vapour and increased expansion in the subsonic region as a result of heat absorption due to the evaporation of liquid droplets behind the shock wave. Therefore, as condensation increases, the shock wave strength decreases. The shock wave shifts upstream along the airfoil surface in the range $375\leqslant T_{\infty }\leqslant 400~\text{K}$ after which it moves downstream in the range $400\leqslant T_{\infty }\leqslant 450~\text{K}$. Behind the airfoil, condensation is reinitiated and $g$ achieves a fully developed value in the far wake.

Figure 11. The $p{-}T$ phase diagram along the NACA0012 airfoil surface for wet steam flow at $\unicode[STIX]{x1D6E9}=0$, $S_{\infty }=1.3$, $M_{\infty }=0.8$ and several $T_{\infty }$.

Figure 12. Variation of $C_{d}$ for steam flow around the NACA0012 airfoil at $S_{\infty }=1.3$, $M_{\infty }=0.8$, $\unicode[STIX]{x1D6E9}=0$ and several $T_{\infty }$.

Figure 12 shows that the pressure drag coefficient ($C_{d}$) increases monotonically with an increase of $T_{\infty }$. The condensation affects the pressure distribution around the airfoil and subsequently, the wave drag in three major ways. First, due to condensation compression, the pressure in the increasing locally supersonic wet flow region increases; this causes the shock wave strength to reduce and wave drag decreases. Second, it shifts the shock wave along the airfoil surface. If the shock wave shifts downstream, drag increases, and if it shifts upstream, drag decreases. Third, as a result of heat absorption due to evaporation of liquid droplets downstream of the shock wave, pressure decreases, which increases the wave drag. In this case, the sum of the second and third effects dominates over the first effect and, thus, an almost linear increase in $C_{d}$ is observed with an increase in $T_{\infty }$ at the prescribed conditions. Also shown, for reference, in figure 12 are $C_{d}$ values for the adiabatic flow cases. Adiabatic flow solutions are computed by imposing a no condensation ($g=0$) condition in the whole domain. The adiabatic $C_{d}$ decreases as $T_{\infty }$ increases at a fixed $S_{\infty }$. This happens because the thermodynamic similarity parameter $K_{G}$ decreases with an increase in $T_{\infty }$, and causes the shock wave to shift upstream on the airfoil surface. The percentage difference between diabatic and adiabatic $C_{d}$ values increases from 11 % at $T_{\infty }=375~\text{K}$ to 50 % at $T_{\infty }=450~\text{K}$.

Zierep & Lin (Reference Zierep and Lin1967) derived a similarity law for the onset of condensation Mach number ($M_{c}$) as it is related to the stagnation relative humidity ($\unicode[STIX]{x1D6F7}_{0}$),

(5.1)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{0}^{a}=\frac{\displaystyle \frac{\unicode[STIX]{x1D6FE}_{e}+1}{2}}{\displaystyle 1+\frac{\unicode[STIX]{x1D6FE}_{e}-1}{2}M_{c}^{2}}.\end{eqnarray}$$

In (5.1), $\unicode[STIX]{x1D6FE}_{e}$ is the effective specific heat ratio of the gas, and exponent $a$ has to be determined from numerical simulations or experiments. In this study, we use $\unicode[STIX]{x1D6FE}_{e}=K_{G}-1$. Note that $K_{G}$ changes primarily with a change in upstream temperature $T_{\infty }$. Figure 13 shows the results of the TSD computed condensation onset Mach number (circles and squares) as a function of $\unicode[STIX]{x1D6F7}_{0}$ for various $T_{\infty }$ at fixed values of $S_{\infty }=1.2$ and $S_{\infty }=1.3$. Also shown for the two fixed values of $S_{\infty }$, are the results of (5.1) (lines). The exponents are found to be $a=0.066$ for $S_{\infty }=1.2$, and $a=0.060$ for $S_{\infty }=1.3$. The exponent $a$ is found to be slightly dependent on $S_{\infty }$, thereby making a separate line for each $S_{\infty }$ value. The TSD results follow the similarity law (5.1) of Zierep & Lin (Reference Zierep and Lin1967) within a computational accuracy of 2 %.

Figure 13. Condensation onset Mach number ($M_{c}$) versus stagnation relative humidity ($\unicode[STIX]{x1D6F7}_{0}$) for steam flows around the NACA0012 airfoil surface at $S_{\infty }=1.2$ and $S_{\infty }=1.3$, $M_{\infty }=0.8$, $\unicode[STIX]{x1D6E9}=0$ and several $T_{\infty }$.

Table 2. Values of $K_{t},K_{G},h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]$, $p_{\infty }$ and the stagnation conditions of fluid for various $S_{\infty }$ at the given conditions.

Next, we analyse the effects of varying the upstream supersaturation ratio ($S_{\infty }$) in the range of 1.2–1.3 at constant values of $M_{\infty }=0.8$ and $T_{\infty }=450~\text{K}$ for wet steam flow around a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) at $\unicode[STIX]{x1D6E9}=0$. Increasing the upstream supersaturation ratio $S_{\infty }$ at a fixed $T_{\infty }$, increases the upstream pressure $p_{\infty }$. Table 2 lists the values of $K_{t}$, $K_{G}$, $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]$, $p_{\infty }$ and the stagnation conditions of upstream fluid for the various flow cases. Note that $n_{c}=18.14,K=1.48$ and $T_{0}=501~\text{K}$ for all the cases. As can be noticed from the values of the various similarity parameters, $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]$ and $K_{G}$ do not vary by much over the range of $S_{\infty }$ studied, whereas $K_{t}$ shows a comparatively larger increase in values. We find that there is only negligible condensation at $S_{\infty }<1.2$. For these flow cases, the critical supersaturation ratio required for initiation of homogeneous nucleation is not attained in any flow region. The increase in $S_{\infty }$ is expected to increase condensation at the prescribed conditions.

Figure 14. Distribution of $-C_{p}$ at the NACA0012 airfoil surface for wet steam flow at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and several $S_{\infty }$.

Figure 15. Distribution of $g$ at the NACA0012 airfoil surface for wet steam flow at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and several $S_{\infty }$.

Figure 16. The $p{-}T$ phase diagram along the surface of the NACA0012 airfoil for wet steam flow at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and several $S_{\infty }$.

Figure 17. Variation of $C_{d}$ for steam flow around the NACA0012 airfoil at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$ and several $S_{\infty }$.

Figures 14 and 15 show the $-C_{p}$ and $g$ distributions at the airfoil surface for the various flow cases. Figure 16 shows a $p{-}T$ phase diagram for a streamline along the airfoil surface for a few selected flow cases. From figure 14, it can be clearly noticed that the condensation compression effect on the flow dynamics increases as $S_{\infty }$ increases. Condensation compression in the supersonic region of the flow and evaporative expansion in the subsonic region behind the shock wave reduces the strength of shock wave. Also, the shock wave shifts upstream on the airfoil surface in the range $1.2\leqslant S_{\infty }\leqslant 1.26$ after which it shifts downstream for $1.26\leqslant S_{\infty }\leqslant 1.3$. From figures 15 and 16, it can be noticed that the condensation region on the airfoil surface increases in axial size and magnitude of $g$ as $S_{\infty }$ increases above 1.2. This is primarily due to the flow experiencing higher local supersaturation ratio ($S$) values as $S_{\infty }$ increases, which results in higher values of nucleation rate $J$. The increase in $K_{t}$ with $S_{\infty }$ also contributes to an increase in condensation as $S_{\infty }$ increases. Also, the point of initiation of condensation moves upstream with the increase of $S_{\infty }$, indicating that the critical supersaturation ratio required for spontaneous homogeneous nucleation is attained relatively upstream on the airfoil surface as $S_{\infty }$ increases. Figure 17 shows the nonlinear increase of $C_{d}$ with increase in $S_{\infty }$. Also shown, for comparison, in figure 17 are the values of $C_{d}$ for the adiabatic flow cases. The adiabatic $C_{d}$ values stay constant at the given conditions, and the percentage difference between the diabatic and adiabatic flows increases as $S_{\infty }$ increases, from 2 % at $S_{\infty }=1.2$ to 50 % at $S_{\infty }=1.3$ due to an increase in condensation. This can be explained by the collective drag-increasing effects of downstream shock wave movement and pressure reduction behind the shock wave (due to evaporative heat absorption) dominating the drag-reducing effect of condensation compression.

Figure 18. Condensation onset Mach number ($M_{c}$) versus stagnation relative humidity ($\unicode[STIX]{x1D6F7}_{0}$) for steam flows around the NACA0012 airfoil surface at $T_{\infty }=400~\text{K}$ and $T_{\infty }=450~\text{K}$, $M_{\infty }=0.8$, $\unicode[STIX]{x1D6E9}=0$ and several $S_{\infty }$.

Figure 18 shows the results of TSD computed condensation onset Mach number (circles and squares) as a function of $\unicode[STIX]{x1D6F7}_{0}$ for various $S_{\infty }$ at fixed values of $T_{\infty }=400~\text{K}$ and $T_{\infty }=450~\text{K}$. Also shown for the two fixed values of $T_{\infty }$ are the results of (5.1) (lines). The exponents are found to be $a=0.062$ for $T_{\infty }=400~\text{K}$, and $a=0.066$ for $T_{\infty }=450~\text{K}$. Note that $K_{G}=2.27$ ($\unicode[STIX]{x1D6FE}_{e}=1.27$) at $T_{\infty }=400~\text{K}$ and $K_{G}=2.20$ ($\unicode[STIX]{x1D6FE}_{e}=1.20$) at $T_{\infty }=450~\text{K}$. Therefore, we have a separate line for each value of $T_{\infty }$. The TSD results show good agreement, within a computational accuracy of 2 %, with the similarity law (5.1) of Zierep & Lin (Reference Zierep and Lin1967).

The effects of independently varying the upstream Mach number ($M_{\infty }$) of wet steam flow around a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) at $\unicode[STIX]{x1D6E9}=0$ are also investigated. Various cases are solved where the upstream conditions are defined as $T_{\infty }=450~\text{K}$, $S_{\infty }=1.27$ and $M_{\infty }$ ranges from 0.78 to 0.83. For all these flow problems, $p_{\infty }=1.18~\text{MPa}$, $n_{c}=18.14$, $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]=2.034$ and $K_{G}=2.203$. Values of $K$, $K_{t}$ and the stagnation conditions of fluid for the various flow cases are provided in table 3. Both $K$ and $K_{t}$ decrease with increase of $M_{\infty }$. Flow is subsonic and there is no condensation for $M_{\infty }<0.76$. It is expected that the supersonic zone over the airfoil increases, the location of the shock wave is pushed downstream and the shock wave becomes stronger with increase of $M_{\infty }$ above 0.76. Thereby, within the subsonic range, the condensation region and size of $g$ also increase with $M_{\infty }$. The transonic similarity parameter $K$ decreases with an increase of $M_{\infty }$. This increases the relative effect of nonlinear terms in (3.29) on the flow and condensation physics.

Figure 19. Distribution of $-C_{p}$ at the NACA0012 airfoil surface for wet steam flow at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $S_{\infty }=1.27$ and several $M_{\infty }$.

Table 3. Values of $K$, $K_{t}$ and the stagnation conditions of fluid for various $M_{\infty }$ at the given conditions.

The distributions of $-C_{p}$ and $g$ at the airfoil surface for several flow cases are shown in figures 19 and 20. A $p{-}T$ diagram for a streamline along the airfoil surface for the various flow cases is shown in figure 21. From figures 20 and 21, it can be observed that the condensation region and size of $g$ increase as $M_{\infty }$ increases. The point of condensation onset and the steady state value of $g$ in the far wake of the airfoil are the same for the various values of $M_{\infty }$. This occurs because the various flow cases have the same values of $n_{c}$, similar $S$ and $\bar{J}_{1}$ values in regions prior to condensation onset. This is also reflected in figure 21, which depicts that the various flow cases follow the same $p{-}T$ phase diagram in regions prior to condensation onset and in the far wake of the airfoil. Figure 19 depicts clearly the increasing effect of condensation compression on the supersonic flow. Figure 22 shows the nonlinear increase of diabatic and adiabatic $C_{d}$ with increasing values of $M_{\infty }$. This is expected as a result of the significant downstream movement and strengthening of the shock wave with an increase in $M_{\infty }$ and at fixed thermodynamic conditions. The difference between the adiabatic and diabatic $C_{d}$ increases until $M_{\infty }\leqslant 0.8$ and reduces as $M_{\infty }$ rises above 0.8. Condensation leads to an increase in wave drag for $M_{\infty }\leqslant 0.82$. However, at higher upstream Mach numbers $M_{\infty }\geqslant 0.83$, condensation results in a decrease in wave drag, primarily due to dominant effects of flow compression.

Figure 20. Distribution of $g$ at the NACA0012 airfoil surface for wet steam flow at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $S_{\infty }=1.27$ and several $M_{\infty }$.

Figure 21. The $p{-}T$ phase diagram along the NACA0012 airfoil surface for wet steam flow at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $S_{\infty }=1.27$ and several $M_{\infty }$.

Figure 22. Variation of $C_{d}$ for steam flow around the NACA0012 airfoil at $\unicode[STIX]{x1D6E9}=0$, $T_{\infty }=450~\text{K}$, $S_{\infty }=1.27$ and several $M_{\infty }$.

Several flow cases are solved to understand the effects of increasing a small angle of attack ($\unicode[STIX]{x1D703}$) on the steady flow and condensation fields of the steam flowing around the airfoil. In these problems, flows of supersaturated steam around a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) with $T_{\infty }=450~\text{K}$, $p_{\infty }=1.12~\text{MPa}$ and $M_{\infty }=0.8$ are studied. For all cases, $T_{0}=501~\text{K}$, $p_{0}=1.75~\text{MPa}$, $\unicode[STIX]{x1D6F7}_{0}=65.2\,\%$, $K=1.48$, $K_{t}=5.49\times 10^{5}$, $K_{G}=2.206$, $n_{c}=18.14$ and $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]=2.026$. We study cases with $\unicode[STIX]{x1D703}=0^{\circ },0.5^{\circ },1^{\circ }$ and $1.5^{\circ }$, for which $\unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D703}/\unicode[STIX]{x1D716}=0,0.073,0.145$ and 0.218, respectively. Figure 23(a,b) presents distributions of $-C_{p}$ on the suction and pressure surfaces of the airfoil, respectively. With increase of $\unicode[STIX]{x1D703}$, the flow on the suction surface accelerates to higher supersonic speeds while the flow on the pressure surface decelerates and even becomes subsonic. Consequently, the shock wave shifts downstream on the suction surface and becomes stronger while the shock wave shifts upstream on the pressure surface and becomes weaker as $\unicode[STIX]{x1D703}$ increases. Shock wave disappears on the pressure surface for $\unicode[STIX]{x1D703}>1^{\circ }$.

Figure 23. Distribution of $-C_{p}$ at (a) suction and (b) pressure surfaces of NACA0012 airfoil for steam flow at $T_{\infty }=450~\text{K}$, $S_{\infty }=1.2$, $M_{\infty }=0.8$ and several $\unicode[STIX]{x1D703}$.

Figure 24. Distribution of $g$ at the suction surface of NACA0012 airfoil for wet steam flow at $T_{\infty }=450~\text{K}$, $S_{\infty }=1.2$, $M_{\infty }=0.8$ and several $\unicode[STIX]{x1D703}$.

Figure 24 shows the distribution of $g$ on the suction surface. As $\unicode[STIX]{x1D703}$ increases, the condensation region expands on the suction surface with higher values of $g$. This happens because of the coupled interaction between flow acceleration on the suction surface and the corresponding pressure and temperature reduction (as noticed in figure 23a,b). This results in the flow attaining higher supersaturation ratio ($S$) values in the supersonic region on the suction surface, which lead to higher nucleation rates ($J$) and hence higher $g$ values. The point of condensation onset also drifts upstream on the airfoil surface with an increase of $\unicode[STIX]{x1D703}$. This is because with an increase of $S$, the flow attains the critical supersaturation ratio value required for initiation of homogeneous nucleation relatively upstream on the airfoil surface. Note that the condensate mass fraction attains the same steady state value (${\sim}0.009$) in the far wake of the airfoil irrespective of $\unicode[STIX]{x1D703}$. In tandem, condensation on the pressure surface becomes negligible with a decrease of velocity and increase of pressure and temperature, and is therefore not shown. Figure 25 shows that the pressure drag coefficient ($C_{d}$) increases monotonically with increase of $\unicode[STIX]{x1D703}$. As $\unicode[STIX]{x1D703}$ increases, the downstream shift in shock wave position, strengthening of the shock wave and reduction in pressure in the supersonic region on the suction surface dominate the increase of wave drag. Also shown for comparison are the values of $C_{d}$ for the adiabatic flow cases. With an increase of angle of attack and consequent increase in condensation, the percentage difference between the $C_{d}$ of diabatic and adiabatic flow cases increases from 4 % at $\unicode[STIX]{x1D703}=0^{\circ }$ to 28 % at $\unicode[STIX]{x1D703}=1.5^{\circ }$. The ratios of lift to wave drag for the flow problems are given in table 4. Also given are values for the adiabatic flow cases. For fixed values of upstream flow conditions, the lift to wave drag ratio increases as angle of attack increases. Compared to the dry flow cases, the lift to drag ratio increases as a result of condensation. As the angle of attack increases and the amount of condensation on the suction surface of the airfoil increases, the jump in the lift to wave drag ratio compared to the corresponding dry flow case increases.

Figure 25. Variation of $C_{d}$ for steam flow around the NACA0012 airfoil at $T_{\infty }=450~\text{K}$, $S_{\infty }=1.2$, $M_{\infty }=0.8$ and several $\unicode[STIX]{x1D703}$.

Table 4. Ratio of lift to drag for various angles of attack at the given conditions.

A few cases are also solved to analyse the effects of airfoil geometry on the flow and condensation fields of wet steam flow over a thin airfoil. Upstream conditions for all the flow problems are $T_{\infty }=450~\text{K}$, $p_{\infty }=1.21~\text{MPa}$ ($S_{\infty }=1.3$), $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$. Stagnation conditions of the upstream fluid are given by $T_{0}=501~\text{K}$, $p_{0}=1.89~\text{MPa}$ and $\unicode[STIX]{x1D6F7}_{0}=70.2\,\%$. All airfoils have a thickness ratio of $0.12$ and a chord length of 0.1 m. The various airfoil geometries selected for analysis are: NACA0012 airfoil, circular arc airfoil, a modified airfoil (see, for shape function, Rusak & Lee (Reference Rusak and Lee2000b)) and an optimum critical airfoil of a sonic arc (see, for shape function, Rusak (Reference Rusak1995)).

The distributions of $-C_{p}$ and $g$ on the various airfoils for a wet steam flow at the given conditions are presented in figures 26 and 27 respectively. For the NACA0012, circular arc and modified airfoils, the effect of condensation compression on the $-C_{p}$ distributions (see figure 26) causes a reduction in shock wave strength due to compression of supersonic flow and expansion of subsonic flow behind shock wave. The condensation region stretches widest for the NACA0012 airfoil with a higher value of maximum $g$. This is due to the curvature of the NACA0012 airfoil which allows the flow to achieve higher speeds, lower pressures and temperatures and higher supersaturation ratio values upon acceleration past the nose region. This also leads to onset of condensation at a point on the NACA0012 airfoil which is relatively upstream as compared to the other airfoils. Due to increased flow acceleration, the flow attains the critical supersaturation ratio required for initiation of homogeneous nucleation far upstream on the NACA0012 airfoil compared to other airfoils. Far downstream of the airfoil, $g$ reaches the same steady state value (${\sim}0.013$) for these airfoils. It is noted that, at the given conditions, the sonic arc does not show any shock wave and condensation along the airfoil surface.

Figure 26. Distribution of $-C_{p}$ at various airfoil surfaces for wet steam flow at $T_{\infty }=450~\text{K}$, $S_{\infty }=1.3$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$.

Figure 27. Distribution of $g$ at various airfoil surfaces for wet steam flow at $T_{\infty }=450~\text{K}$, $S_{\infty }=1.3$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$.

The $C_{d}$ values of the dry and wet steam flow around the various airfoils at the prescribed conditions are provided in table 5. Wave drag on the optimum critical sonic arc airfoil is found to be zero in both cases. Condensation leads to an increase in drag for the other airfoil geometries. The rise in drag for the circular arc is 0.0021, for the modified airfoil is 0.0015 and for the NACA0012 airfoil is 0.0027. The rise in wave drag is primarily due to the shift of shock wave position and reduction of pressure downstream of the shock wave.

Table 5. Values of $C_{d}$ for the various airfoils at the given conditions.

It is known that, due to the mass flow constraint of internal flows through nozzles and channels by walls (Schnerr & Dohrmann Reference Schnerr and Dohrmann1990, Reference Schnerr and Dohrmann1994), increasing the amount of heat addition due to condensation results in the initiation of self-excited high frequency oscillations in the flow. Identification of a similar stability limit in external flows remains a problem of interest. The current TSD theory helps in providing an insight into the limit of the amount of condensation heat release to the steam flow beyond which a steady state solution does not exist. The present computations are performed where steam at $M_{\infty }=0.8$ flows around a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) at $\unicode[STIX]{x1D6E9}=0$. Upstream supersaturation ratio $S_{\infty }$ is increased at a fixed $T_{\infty }$ until the computations show oscillations, fail to converge and provide a steady state solution. Table 6 lists the values of maximum upstream supersaturation ratio $S_{\infty }$ for which steady state solution at a particular $T_{\infty }$ exists according to the current TSD model. As $T_{\infty }$ is increased at fixed values of $M_{\infty }$ and airfoil geometry and angle of attack, the steady state limiting $S_{\infty }$ decreases. Increasing the heat addition to the steam flow beyond a certain amount, for example, beyond a maximum condensate mass fraction generation on the airfoil surface of 3 %, resulted in the numerical algorithm producing oscillations, and not converging to a steady state.

Table 6. Values of maximum $S_{\infty }$ for various $T_{\infty }$ at the given conditions.

Virk & Rusak (Reference Virk and Rusak2019) derived a transonic small-disturbance theory to describe the physics of pure steam flow with non-equilibrium and homogeneous condensation where thermodynamics of steam flow was modelled by the perfect-gas law. Comparisons of numerical results of the current TSD model which takes into account the real-gas effects with the perfect-gas model may provide meaningful insights into the sensitivity of the TSD numerical results to thermodynamic modelling of steam.

First, a flow problem is solved using the two TSD models where pure steam with free-stream conditions defined by $M_{\infty }=0.8$, $T_{\infty }=375~\text{K}$ and $S_{\infty }=0.8$ ($p_{\infty }=0.87~\text{bar}$) flows around a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) at zero angle of attack ($\unicode[STIX]{x1D6E9}=0$). The stagnation properties of the upstream fluid are: $p_{0}=1.23$ bar, $T_{0}=413~\text{K}$ and $\unicode[STIX]{x1D6F7}_{0}=34.3\,\%$. Similarity parameters for the flow problem are: $K=1.48$, $K_{t}=3.16\times 10^{4}$, $K_{G}=2.293$, $n_{c}=51.49$ and $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]=2.99$. No condensation is observed in any flow region for both flow cases because the low upstream supersaturation ratio does not allow the flow to reach the critical supersaturation ratio required for initiation of homogeneous nucleation anywhere in the flow. The value of $K_{G}=2.29$ for a van der Waals gas is close to the perfect gas $K_{G}=\unicode[STIX]{x1D6FE}_{v}+1=2.33$, and the thermodynamics of steam flow in these flow conditions can be adequately described by the perfect-gas law. The solution of the two models nearly overlap in most of the flow region.

The numerical results of the van der Waals gas TSD model are also compared to perfect-gas TSD model for a condensing pure steam flow problem. For this, a pure steam flow around a NACA0012 airfoil ($c=0.1~\text{m}$, $\unicode[STIX]{x1D716}=0.12$) at zero angle of attack ($\unicode[STIX]{x1D6E9}=0$) and at free-stream conditions defined by $M_{\infty }=0.8$, $T_{\infty }=375~\text{K}$ and $S_{\infty }=1.3$ ($p_{\infty }=1.41~\text{bar}$) is considered. The stagnation properties of fluid are $p_{0}=2.24~\text{bar}$, $T_{0}=426~\text{K}$ and $\unicode[STIX]{x1D6F7}_{0}=43.3\,\%$. Similarity parameters for the flow problem are $K=1.48$, $K_{t}=5.14\times 10^{4}$, $K_{G}=2.291$, $n_{c}=51.49$ and $h_{fg}(T_{\infty })/[K_{3}C_{vv}T_{\infty }(1+K_{r})]=3$.

Figure 28. Distribution of $-C_{p}$ at the NACA0012 airfoil surface for wet steam flow at $T_{\infty }=375~\text{K}$, $S_{\infty }=1.3$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$ according to the perfect-gas and van der Waals gas TSD models.

Figure 29. Distribution of $g$ at the NACA0012 airfoil surface for wet steam flow at $T_{\infty }=375~\text{K}$, $S_{\infty }=1.3$, $M_{\infty }=0.8$ and $\unicode[STIX]{x1D6E9}=0$ according to the perfect-gas and van der Waals gas TSD models.

Figures 28 and 29 show the distributions of the pressure coefficient and condensate mass fraction along the airfoil surface, respectively. Figure 29 shows that the perfect-gas TSD model predicts higher condensation and the condensation initiation occurs relatively upstream in the flow compared to van der Waals gas TSD model. The shock wave for the van der Waals gas model is located upstream and does not show condensation compression effects as significant as in the perfect-gas model (see figure 28). Due to significant condensation in the supersonic region ahead of shock wave for the perfect gas, the shock wave also drifts downstream towards the trailing edge as the pressure is increased in the local supersonic region. The condensation compression ahead of shock wave and evaporation heat release behind the shock wave, together reduce the shock wave strength considerably for the perfect-gas case relative to the van der Waals gas solution. Thus, it can be concluded that the predictions according to TSD models are sensitive to the thermodynamic modelling of the gas for flows at high upstream pressures ($p_{\infty }>2~\text{bar}$) and temperatures ($T_{\infty }>400~\text{K}$). Moreover, even for flows at low temperatures, which involve condensation initiation, thermodynamic sensitivity is significant, although without condensation there is not much difference between the predictions of the two TSD models. Real-gas behaviour needs to be considered in describing the flow thermodynamics under such conditions to improve modelling of the physical behaviour according to TSD theory.