Who First Solved the Continuity and Momentum Equation for a Flat Plate

Boundary-Layer Equations

TUNCER CEBECI , in Analysis of Turbulent Flows, 2004

3.6.1 Momentum Integral Equation

The momentum integral equation for a two-dimensional steady compressible flow can be obtained by integration from the boundary-layer Equations (3.3.5) and (3.3.6b). ⁴ If we multiply Eq. (3.3.5) by (u _e - u), multiply Eq. (3.3.6b) by −1, add and subtract qu(du _e/dx) from Eq. (3.3.6b), and add the resulting continuity and momentum equations, we can arrange the resulting expression in the form

$\begin{array}{l}\frac{\partial }{\partial x}\text{[}\varrho u\left(u_{\text{e}}\text{-}u\text{)|+}\frac{\partial }{\partial y}\text{[}\varrho v\right(u_e\text{-}u\text{)]+}\frac{d{u}_{\text{e}}}{dx}({\varrho }_{\text{e}}{u}_{\text{e}}\text{-}\varrho \text{u)}\\ \text{=-}\frac{\partial }{\partial y}\left(\mu \frac{\partial u}{\partial y}\text{-}\varrho \overline{u^{\prime }{v}^{\prime }}\right)\text{.}\end{array}$

Nondimensionalizing and integrating with respect to y from zero to infinity, we get

(3.6.1) $\begin{array}{l}\frac{d}{dx}\text{[}{\varrho }_{\text{e}}{u}_{\text{e}}^{\text{2}}{{{\displaystyle \int }}^{}}_{\text{0}}^{\infty }\frac{\varrho u}{{\varrho }_e{u}_e}\text{(1-}\frac{u}{u_e}\left)dy\text{]+}\frac{d{u}_e}{dx}{\varrho }_{\text{e}}{u}_{\text{e}}\text{[}{{{\displaystyle \int }}^{}}_{\text{0}}^{\infty }\text{(1-}\frac{\varrho u}{{\varrho }_e{\text{u}}_{\text{e}}}\right)dy\text{]-}{\varrho }_{\text{w}}{v}_{\text{w}}{u}_{\text{e}}\\ ={\left(\mu \frac{\partial u}{\partial y}\right)}_{\text{w}}\equiv {\tau }_{\text{w}},\end{array}$

since (∂u/∂y) and $\overline{u^{\prime }{v}^{\prime }}$ → 0 as y → ∞ and since $\overline{u^{\prime }{v}^{\prime }}$ → 0 as y → 0. It is more convenient to express Eq. (3.6.1) in terms of boundary-layer thicknesses δ^* and θ. Equation (3.6.1) then becomes

$\frac{d}{dx}\left({\varrho }_{\text{e}}{\text{u}}_{\text{e}}^{\text{2}}\theta \right)+{\varrho }_{\text{e}}{u}_{\text{e}}{\delta }^{\text{*}}\frac{d{u}_{\text{e}}}{dx}\text{-}{\varrho }_{\text{w}}{v}_{\text{w}}{u}_{\text{e}}={\tau }_{\text{w}},$

or, in nondimensional form,

(3.6.2) $\frac{d\theta }{dx}\text{+}\frac{\theta }{u_{\text{e}}}(H\text{+2)}\frac{d{u}_{\text{e}}}{dx}\text{+}\frac{\theta }{{\varrho }_e}\frac{d{\varrho }_{\text{e}}}{dx}\text{-}\frac{{\varrho }_{\text{w}}}{{\varrho }_e}\frac{v_{\text{w}}}{u_{\text{e}}}=\frac{{\tau }_{\text{w}}}{{\varrho }_{\text{e}}{u}_{\text{e}}^{\text{2}}},$

where H denotes the ratio δ^*/θ, which is known as the shape factor. For an ideal gas undergoing an isentropic process, we can write

(3.6.3) $\frac{\text{1}}{{\varrho }_e}\frac{d{\varrho }_e}{dx}\text{=-}\frac{M_{\text{e}}^{\text{2}}}{u_{\text{e}}}\frac{d{\text{u}}_e}{dx}.$

Substituting from that equation into Eq. (3.6.2) and rearranging, we obtain the momentum integral equation of the boundary layer for a two-dimensional compressible flow:

(3.6.4) $\frac{d\theta }{dx}\text{+}\frac{\theta }{u_{\text{e}}}\left(H\text{+2-}{M}_{\text{e}}^{\text{2}}\right)\frac{d{\text{u}}_{\text{e}}}{dx}\text{-}\frac{{\varrho }_{\text{w}}}{{\varrho }_e}\frac{v_{\text{w}}}{u_{\text{e}}}=\frac{{\tau }_{\text{w}}}{{\varrho }_e{\text{u}}_{\text{e}}^{\text{2}}}\equiv \frac{c_{\text{f}}}{\text{2}},$

where c _f is the local skin-friction coefficient. Note that in the case of zero mass transfer the normal velocity component at the wall v _w is zero. Then Eq. (3.6.4) becomes

(3.6.5) $\frac{d\theta }{dx}\text{+}\frac{\theta }{u_{\text{e}}}\left(H\text{+2-}{M}_{\text{e}}^{\text{2}}\right)\frac{d{u}_e}{dx}=\frac{c_{\text{f}}}{\text{2}}.$

For an incompressible flow with no mass transfer, that equation reduces to

(3.6.6) $\frac{d\theta }{dx}\text{+}\frac{\theta }{u_{\text{e}}}(H\text{+2)}\frac{d{u}_{\text{e}}}{dx}=\frac{c_{\text{f}}}{\text{2}}.$

Equations (3.6.4)–(3.6.6) are also known as the first momentum integral equations. They are applicable to both laminar and turbulent boundary layers.

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Boundary-Layer Equations

Tuncer Cebeci , in Analysis of Turbulent Flows with Computer Programs (Third Edition), 2013

3.6.1 Momentum Integral Equation

The momentum integral equation for a two-dimensional steady compressible flow can be obtained by integration from the boundary-layer equations (3.3.5) and (3.3.6b). ⁴ If we multiply Eq. (3.3.5) by (u _e–u), multiply Eq. (3.3.6b) by –1, add and subtract $\varrho u(d{u}_{\mathrm{e}}/dx)$ from Eq. (3.3.6b), and add the resulting continuity and momentum equations, we can arrange the resulting expression in the form

$\begin{array}{l}\frac{\partial }{\partial x}\left[\varrho u\right(u_{\mathrm{e}}-u\left)\right]+\frac{\partial }{\partial y}\left[\varrho \upsilon \right(u_{\mathrm{e}}-u\left)\right]+\frac{d{u}_e}{dx}({\varrho }_{\mathrm{e}}{u}_{\mathrm{e}}-\varrho u)=-\frac{\partial }{\partial y}\left(\mu \frac{\partial u}{\partial y}-\varrho \overline{u\mathit{\text{'}}\upsilon \mathit{\text{'}}}\right).\end{array}$

Nondimensionalizing and integrating with respect to y from zero to infinity, we get

(3.6.1) $\begin{array}{l}\frac{d}{dx}\left[{\varrho }_{\text{e}}{u}_{\text{e}}^2{\int }_0^{\mathrm{\infty }}\frac{\varrho u}{{\varrho }_{\text{e}}{u}_{\text{e}}}\left(1-\frac{u}{u_{\text{e}}}\right)dy\right]+\frac{d{u}_{\text{e}}}{dx}{\varrho }_{\text{e}}{u}_{\text{e}}\left[{\int }_0^{\mathrm{\infty }}\left(1-\frac{\varrho u}{{\varrho }_{\text{e}}{u}_{\text{e}}}\right)dy\right]-{\varrho }_{\text{w}}{\upsilon }_{\text{w}}{u}_{\text{e}}\\ ={\left(\mu \frac{\partial u}{\partial y}\right)}_{\text{w}}\equiv {\tau }_{\text{w}},\end{array}$

since ( $\partial u/\partial y$ ) and $\overline{u^{\prime }{\upsilon }^{\prime }}$ → 0 as y → ∞ and since $\overline{u^{\prime }{\upsilon }^{\prime }}$ → 0 as y → 0. It is more convenient to express Eq. (3.6.1) in terms of boundary-layer thicknesses δ^∗ and θ. Equation (3.6.1) then becomes

$\frac{d}{dx}\left({\varrho }_{\text{e}}{u}_{\text{e}}^2\theta \right)+{\varrho }_{\text{e}}{u}_{\text{e}}\delta \ast \frac{d{u}_{\text{e}}}{dx}-{\varrho }_{\text{w}}{\upsilon }_{\text{w}}{u}_{\text{e}}={\tau }_{\text{w}},$

or, in nondimensional form,

(3.6.2) $\frac{d\theta }{dx}+\frac{\theta }{u_{\text{e}}}(H+2)\frac{d{u}_{\text{e}}}{dx}+\frac{\theta }{{\varrho }_{\text{e}}}\frac{d{\varrho }_{\text{e}}}{dx}-\frac{{\varrho }_{\text{w}}}{{\varrho }_{\text{e}}}\frac{{\upsilon }_{\text{w}}}{u_{\text{e}}}=\frac{{\tau }_{\text{w}}}{{\varrho }_{\text{e}}{u}_{\text{e}}^2},$

where H denotes the ratio δ ^∗/θ, which is known as the shape factor. For an ideal gas undergoing an isentropic process, we can write

(3.6.3) $\frac{1}{{\varrho }_{\text{e}}}\frac{d{\varrho }_{\text{e}}}{dx}=-\frac{M_{\text{e}}^2}{u_{\text{e}}}\frac{d{u}_{\text{e}}}{dx}.$

Substituting from that equation into Eq. (3.6.2) and rearranging, we obtain the momentum integral equation of the boundary layer for a two-dimensional compressible flow:

(3.6.4) $\frac{d\theta }{dx}+\frac{\theta }{u_{\text{e}}}(H+2-M_{\text{e}}^2)\frac{d{u}_{\text{e}}}{dx}-\frac{{\varrho }_{\text{w}}}{d_{\text{e}}}\frac{{\upsilon }_{\text{w}}}{u_{\text{e}}}=\frac{{\tau }_{\text{w}}}{{\varrho }_{\text{e}}{u}_{\text{e}}^2}\equiv \frac{c_{\text{f}}}{2},$

where c_f is the local skin-friction coefficient. Note that in the case of zero mass transfer the normal velocity component at the wall υ_w is zero. Then Eq. (3.6.4) becomes

(3.6.5) $\frac{d\theta }{dx}+\frac{\theta }{u_{\text{e}}}(H+2-M_{\text{e}}^2)\frac{d{u}_{\text{e}}}{dx}=\frac{c_{\text{f}}}{2}.$

For an incompressible flow with no mass transfer, that equation reduces to

(3.6.6) $\frac{d\theta }{dx}+\frac{\theta }{u_{\text{e}}}(H+2)\frac{d{u}_{\text{e}}}{dx}=\frac{c_{\text{f}}}{2}.$

Equations (3.6.4)–(3.6.6) are also known as the first momentum integral equations. They are applicable to both laminar and turbulent boundary layers.

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GENERALIZED GOVERNING EQUATIONS IN MULTIPHASE SYSTEMS: LOCAL INSTANCE FORMULATIONS

Amir Faghri , Yuwen Zhang , in Transport Phenomena in Multiphase Systems, 2006

Momentum Balance

The momentum integral equation for a control volume containing two phases separated by an interface is given by eq. (3.13) with Π = 2. It can be rewritten as

(3.168) $\begin{array}{l}{\displaystyle \sum _{k=1}^2\left[{\displaystyle {\int }_{{\text{V}}_k\left(t\right)}\left({\displaystyle \sum _{i=1}^N{\rho }_{k,i}{\text{X}}_{k,i}}\right)dV}+{\displaystyle {\int }_{A_k\left(t\right)+A_I\left(t\right)}{{\tau }^{\prime }}_{k,rel}.{\text{n}}_{k}dA}\right]}-{\displaystyle {\int }_{A_I\left(t\right)}\left[{\displaystyle \sum _{k=1}^2{{\tau }^{\prime }}_{k,rel}.{\text{n}}_k}\right]dA}\\ ={\displaystyle \sum _{k=1}^2\left[\frac{\partial }{\partial t}{\displaystyle {\int }_{V_k\left(t\right)}{\rho }_k{A}_{k,rel}}dV+{\displaystyle {\int }_{{\text{V}}_k\left(t\right)+A_I\left(t\right)}{\rho }_k\left({\text{V}}_{k,rel}.{\text{n}}_k\right)}{\text{V}}_{k,rel}dA\right]}-{\displaystyle {\int }_{A_I(t)}\left[{\displaystyle \sum _{k=1}^2{\rho }_k\left({\text{V}}_{k,rel}.{\text{n}}_{k,rel}\right)}\right]}dA\end{array}$

The surface integrals over A _k (t) + A _I (t) in eq. (3.168) can be converted to volume integrals over V _k (t) using eq. (3.34):

(3.169) $\begin{array}{l}{\displaystyle \sum _{k=\text{l}}^2{\displaystyle {\int }_{V_k\left(t\right)}\left[\nabla \cdot {{\tau }^{\prime }}_k+{\displaystyle \sum _{i=\text{l}}^N{\rho }_{k,i}{\text{X}}_{k,i}-\frac{\partial }{\partial t}\left({\rho }_k{\text{V}}_k\right)-\nabla \cdot {\rho }_k{\text{V}}_{k,rel}{\text{V}}_k}\right]}}dV\\ -{\displaystyle {\int }_{A_I\left(t\right)}\left[{\displaystyle \sum _{k=\text{l}}^2{{\tau }^{\prime }}_{k,rel}\cdot {\text{n}}_k-{\rho }_k\left({\text{V}}_{k,rel}\cdot {\text{n}}_k\right){\text{V}}_{k,rel}}\right]}dA\end{array}$

Substituting eq. (3.52) into eq. (3.169), one obtains

(3.170) ${\displaystyle {\int }_{A_I\left(t\right)}\left[{\displaystyle \sum _{k=\text{l}}^2{{\tau }^{\prime }}_{k,rel}\cdot {\text{n}}_k-{\rho }_k\left({\text{V}}_{k,rel}\cdot {\text{n}}_k\right){\text{V}}_{k,rel}}\right]}dA=0$

As stated above, the only condition that assures the general validity of eq. (3.170) is that the integrand equals zero, so

(3.171) ${{\tau }^{\prime }}_1\cdot {\text{n}}_{\text{1}}+{{\tau }^{\prime }}_2\cdot {\text{n}}_{\text{2}}={\rho }_1\left({\text{V}}_{\text{1,}rel}\cdot {\text{n}}_{\text{1}}\right){\text{V}}_{\text{1,}rel}+{\rho }_2\left({\text{V}}_{\text{2}}{}_{,rel}\cdot {\text{n}}_{\text{2}}\right){\text{V}}_{\text{2,}rel}$

If the reference frame velocity equals the velocity of the interface, eq. (3.171) becomes

(3.172) ${{\tau }^{\prime }}_1\cdot {\text{n}}_{\text{1}}+{{\tau }^{\prime }}_2\cdot {\text{n}}_{\text{2}}={\rho }_1\left[\left({\text{V}}_{\text{1}}-{\text{V}}_I\right)\cdot {\text{n}}_{\text{1}}\right]\left({\text{V}}_{\text{1}}-{\text{V}}_I\right)\text{+}{\rho }_2\left[\left({\text{V}}_{\text{2}}-{\text{V}}_I\right)\cdot {\text{n}}_{\text{2}}\right]\left({\text{V}}_{\text{2}}-{\text{V}}_I\right)$

Substituting eqs. (3.163) and (3.165) into eq. (3.172), the momentum balance becomes

(3.173) $\left({{\tau }^{\prime }}_1-{{\tau }^{\prime }}_2\right)\cdot {\text{n}}_1={{\dot{m}}^{‴}}_1\left({\text{V}}_{\text{1}}-{\text{V}}_{\text{2}}\right)$

Substituting eq. (3.55) into eq. (3.173), one obtains

(3.174) $\left(p_2-p_1\right){\text{n}}_{\text{1}}+\left({\tau }_1-{\tau }_2\right)\cdot {\text{n}}_{\text{1}}={\dot{m}}^{″}\left({\text{V}}_{\text{1}}-{\text{V}}_{\text{2}}\right)$

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Short Cut Methods

Tuncer Cebeci , in Analysis of Turbulent Flows with Computer Programs (Third Edition), 2013

Head's Method

The momentum integral equation

(3.6.6) $\frac{d\theta }{dx}+\frac{\theta }{u_e}\frac{d{u}_e}{dx}(H+2)=\frac{c_f}{2}$

contains the three unknowns θ, H, and c_f , and assumed relationships between these integral parameters are required. There are several approaches to the achievement of this objective. One approach that we shall consider here adopts the notion that a turbulent boundary layer grows by a process of "entrainment" of nonturbulent fluid at the outer edge and into the turbulent region. It was first used by Head [17], who assumed that the mean-velocity component normal to the edge of the boundary layer (which is known as the entrainment velocity υ_E ) depends only on the mean-velocity profile, specifically on H. He assumed that the dimensionless entrainment velocity υ_E /u_e is given by

(7.3.1) $\frac{{\upsilon }_E}{u_e}\equiv \frac{1}{u_e}\frac{d}{dx}{\int }_0^{\delta }udy=\frac{1}{u_e}\frac{d}{dx}{u}_e(\delta -{\delta }^{\ast })=F\left(H_1\right),$

where we have used the definition of δ ^∗ for two-dimensional incompressible flows. If we define

(7.3.2) $H_1=\frac{\delta -{\delta }^{\ast }}{\theta },$

then the right-hand equality in Eq. (7.3.1) can be written as

(7.3.3) $\frac{d}{dx}\left(u_e\theta H_1\right)=u_{e}F.$

Head also assumed that H ₁ is related to the shape factor H by

(7.3.4) $H_1=G\left(H\right).$

The functions F and G were determined from experiment, and a best fit to several sets of experimental data showed that they can be approximated by

(7.3.5) $F=0.0306{\left(H_1-3.0\right)}^{-0.6169},$

(7.3.6) $G=\{\begin{array}{ll}0.8234{(H-1.1)}^{-1.287}+3.3\hfill & H\le 1.6,\hfill \\ 1.5501{(H-0.6778)}^{-3.064}+3.3\hfill & H\ge 1.6.\hfill \end{array}$

With F and G defined by Eqs. (7.3.5) and (7.3.6), Eq. (7.3.3) provides a relationship between θ and H. Another equation relating c_f to θ and/or H is needed, and Head used the semiempirical skin-friction law given by Ludwieg and Tillmann [18],

(7.3.7) $c_f=0.246\times {10}^{-0.678H}{R}_{\theta }^{-0.268},$

where R_θ   = u_eθ/ν. The system [Eqs. (3.6.6) and (7.3.1)–(7.3.7)], which includes two ordinary differential equations, can be solved numerically for a specified external velocity distribution to obtain the boundary-layer development on a two-dimensional body with a smooth surface [19]. To start the calculations, say x  = x ₀, we note that initial values of two of the three quantities θ, H and c_f must be specified, with the third following from Eq. (7.3.7). When turbulent-flow calculations follow laminar calculations for a boundary layer on the same surface, Head's method is often started by assuming continuity of momentum thickness θ and taking the initial value of H to be 1.4, an approximate value corresponding to flat-plate flow.

This model, like most integral methods, uses a given value of the shape factor H as the criterion for separation. [Equation (7.3.7) predicts c_f to be zero only if H tends to infinity]. It is not possible to give an exact value of H corresponding to separation, and values between the lower and upper limits of H makes little difference in locating the separation point since the shape factor increases rapidly close to separation.

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Short Cut Methods

TUNCER CEBECI , in Analysis of Turbulent Flows, 2004

Head's Method

The momentum integral equation

(3.6.6) $\frac{d\theta }{dx}\text{+}\frac{\theta }{u_{\text{e}}}\frac{d{u}_e}{dx}(H+2)=\frac{c_f}{2}$

contains the three unknowns θ, H, and c _f , and assumed relationships between these integral parameters are required. There are several approaches to the achievement of this objective. One approach that we shall consider here adopts the notion that a turbulent boundary layer grows by a process of "entrainment" of nonturbulent fluid at the outer edge and into the turbulent region. It was first used by Head [17], who assumed that the mean-velocity component normal to the edge of the boundary layer (which is known as the entrainment velocity v _E )depends only on the mean-velocity profile, specifically on H. He assumed that the dimensionless entrainment velocity V _E /u _e is given by

(7.3.1) $\frac{v_E}{u_e}\equiv \frac{\text{1}}{u_{\text{e}}}\frac{d}{dx}{{{\displaystyle \int }}^{}}_{\text{0}}^{\delta }udy=\frac{\text{1}}{u_{\text{e}}}\frac{d}{dx}\text{[}{u}_e\left(\delta \text{-}{\delta }^{\text{*}}\right)]=F(H_{\text{1}}\text{),}$

where we have used the definition of δ^* for two-dimensional incompressible flows. If we define

(7.3.2) $H_{\text{1}}=\frac{\delta \text{-}{\delta }^{\text{*}}}{\theta },$

then the right-hand equality in Eq. (7.3.1) can be written as

(7.3.3) $\frac{d}{dx}{\text{(u}}_{\text{e}}\theta H_{\text{1}})=u_{e}F.$

Head also assumed that H ₁ is related to the shape factor H by

(7.3.4) $H_{\text{1}}=G\left(H\right).$

The functions F and G were determined from experiment, and a best fit to several sets of experimental data showed that they can be approximated by

(7.3.5) $F=0\text{.0306(}{H}_{\text{1}}\text{-3}{\text{.0)}}^{\text{-0}\text{.6169}},$

(7.3.6) $G=\{\begin{array}{ll}\text{0}\text{.8234(}H\text{-1}{\text{.1)}}^{\text{-1}\text{.287}}\text{+3}\text{.3}\hfill & H\le \text{1}\text{.6,}\hfill \\ \text{1}\text{.5501(}H\text{-0}{\text{.6778)}}^{\text{-3}\text{.064}}\text{+3}\text{.3}\hfill & H\ge \text{1}\text{.6}\text{.}\hfill \end{array}$

With F and G defined by Eqs. (7.3.5) and (7.3.6), Eq. (7.3.3) provides a relationship between θ and H. Another equation relating c _f to θ and/or H is needed, and Head used the semiempirical skin-friction law given by Ludwieg and Tillmann [18],

(7.3.7) $c_f=0{\text{.246×10}}^{\text{-0}\text{.678}H}{R}_{\theta }^{\text{-0}\text{.268}},$

where R _θ = u _e θ/v. The system [Eqs. (3.6.6) and (7.3.1)–(7.3.7)], which includes two ordinary differential equations, can be solved numerically for a specified external velocity distribution to obtain the boundary-layer development on a two-dimensional body with a smooth surface [19]. To start the calculations, say x = x ₀, we note that initial values of two of the three quantities θ, H and c _f must be specified, with the third following from Eq. (7.3.7). When turbulent-flow calculations follow laminar calculations for a boundary layer on the same surface, Head's method is often started by assuming continuity of momentum thickness θ and taking the initial value of H to be 1.4, an approximate value corresponding to flat-plate flow.

This model, like most integral methods, uses a given value of the shape factor H as the criterion for separation. [Equation (7.3.7) predicts c _f to be zero only if H tends to infinity]. It is not possible to give an exact value of H corresponding to separation, and values between the lower and upper limits of H makes little difference in locating the separation point since the shape factor increases rapidly close to separation.

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Computational Fluid Dynamics

E.L. Houghton , ... Daniel T. Valentine , in Aerodynamics for Engineering Students (Seventh Edition), 2017

9.1.1 Methods Based on the Momentum-Integral Equation

In the general case with an external pressure gradient, the momentum-integral equation must be solved numerically. There are a number of ways this can be done. One, for laminar boundary layers, is the approximate expressions with Eq. (3.75) or Eq.(3.193) rewritten as

(9.1) $\frac{d\theta }{dx}=\frac{C_f}{2}-\frac{V_{\mathrm{s}}}{U_e}-\frac{\theta }{U_e}\frac{d{U}_e}{dx}(H+2)$

$d\theta /dx$ can be related to $d\delta /dx$ :

$\begin{array}{ccc}\hfill \frac{d\theta }{dx}& \hfill =\hfill & \frac{d(\delta I)}{dx}=I\frac{d\delta }{\mathrm{dss}x}+\delta \frac{dI}{dx}=I\frac{d\delta }{dx}+\delta \frac{dI}{d\mathrm{\Lambda }}\frac{d\mathrm{\Lambda }}{d\delta }\frac{d\delta }{dx}\hfill \\ \hfill & \hfill =\hfill & (I+\delta \frac{dI}{d\mathrm{\Lambda }}\frac{d\mathrm{\Lambda }}{d\delta })\frac{d\delta }{dx}\hfill \end{array}$

It follows from Eqs.(3.85),(3.94),(3.95) and(3.96), respectively, that

$\begin{array}{ccc}\hfill \delta \frac{d\mathrm{\Lambda }}{d\delta }& \hfill =\hfill & \frac{2{\delta }^2}{\nu }\frac{d{U}_e}{dx}=2\mathrm{\Lambda }\hfill \\ \hfill \frac{dI}{d\mathrm{\Lambda }}& \hfill =\hfill & -\frac{1}{63}(\frac{1}{15}+\frac{\mathrm{\Lambda }}{72})\hfill \end{array}$

So

(9.2) $\frac{d\theta }{dx}=F_1(\mathrm{\Lambda })\frac{d\delta }{dx}$

where

$F_1(\mathrm{\Lambda })=I-\frac{2\mathrm{\Lambda }}{63}(\frac{1}{15}+\frac{\mathrm{\Lambda }}{72})$

Thus, Eq.(9.1) can be readily converted into an equation for $\mathrm{d}\delta /\mathrm{d}x$ by dividing both sides by $F_1(\mathrm{\Lambda })$ . The problem is that it follows from Eq.(3.96) that

$C_f\propto \frac{1}{U_e\delta }$

Thus, in cases where either $\delta =0$ or $U_e=0$ at $x=0$ , the initial value of $C_f$ , and therefore the right-hand side of Eq.(9.1) is infinite. These two cases are in fact the two most common in practice. The former corresponds to sharp leading edges; the latter, to blunt ones. To deal with the problem identified, both sides of Eq.(9.1) are multiplied by $2U_e\delta /v$ , whereby, using Eq.(9.2), it becomes

$F_1(\mathrm{\Lambda })\frac{U_e}{v}\frac{d{\delta }^2}{dx}=F_2(\mathrm{\Lambda })-\frac{2V_{\mathrm{s}}\delta }{v}$

where Eqs.(3.94) to(3.96) give

$F_2(\mathrm{\Lambda })=4+\frac{\mathrm{\Lambda }}{3}-\frac{3}{10}\mathrm{\Lambda }+\frac{{\mathrm{\Lambda }}^2}{60}+\frac{4}{63}(\frac{37}{5}-\frac{\mathrm{\Lambda }}{15}-\frac{{\mathrm{\Lambda }}^2}{144})$

To obtain the final form of Eq.(9.1) for computational purposes, $F_1(\mathrm{\Lambda }){\delta }^{\mathrm{2}}\mathrm{d}{U}_e/\mathrm{d}x$ is added to both sides, which are then divided by $F_1(\mathrm{\Lambda })$ ; thus

(9.3) $\frac{dZ}{dx}=\frac{F_2(\mathrm{\Lambda })}{F_1(\mathrm{\Lambda })}+\mathrm{\Lambda }-\frac{2}{F_1(\mathrm{\Lambda })}\frac{V_{\mathrm{s}}\delta }{v}$

where the dependent variable changes from δ to $Z={\delta }^{\mathrm{2}}{U}_e/v$ . In the usual case when $V_{\mathrm{s}}=0$ , the right-hand side of Eq.(9.3) is purely a function of Λ. Note that

$\mathrm{\Lambda }=\frac{Z}{U_e}\frac{d{U}_e}{dx}\text{and}\delta =\sqrt{\frac{Zv}{U_e}}$

Since $U_e$ is a prescribed function of x, this allows both Λ and δ to be obtained from a value of Z. The other quantities of interest can be obtained from Eqs.(3.94) to(3.96).

With the momentum-integral equation in the form of Eq.(9.3) it is suitable for the direct application of standard methods for numerical integration of ordinary differential equations [77]. We recommend that the fourth-order Runge-Kutta method be used with an adaptive stepsize control; the advantage of this is that small steps are chosen in regions of rapid change (e.g., near the leading edge), while larger steps are taken elsewhere.

To begin the calculation, it is necessary to supply initial values for Z and Λ at, say, $x=0$ . For a sharp leading edge, $\delta =0$ , giving $Z=\mathrm{\Lambda }=0$ at $x=0$ , whereas for a round leading edge, $U_e=0$ , and $Z=0$ , but $\mathrm{\Lambda }=7.052$ (see Example 3.8). This value of Λ should be used to evaluate the right-hand side of Eq.(9.3) at $x=0$ .

Boundary-layer computations using the method just described were carried out for a circular cylinder of radius l m in air flowing at 20 m/s ( $v=1.5\times {10}^{-5}{\text{ m}}^2/\text{s}$ ). In Fig. 9.1, the computed values of Λ and momentum thickness are plotted against the angle around the cylinder's surface measured from the fore stagnation point. A fourth-order Runge-Kutta integration scheme was used with 200 fixed steps between $x=0$ and $x=3$ , which gave acceptable accuracy. According to this approximate calculation, the separation point, corresponding to $\mathrm{\Lambda }=-12$ , occurs at 106.7 degrees. This should be compared with the accurately computed value of 104.5 degrees obtained with the differential form of the equations of motion. It can be seen that the Pohlhausen [78] method gives reasonably acceptable results when compared with more accurate methods. In point of fact, neither value given for the separation point is close to the actual value found experimentally for a laminar boundary layer. Experimentally, separation occurs just ahead of the apex of the cylinder. The reason for the large discrepancy between theoretical and observed separation points is that the large wake substantially alters the flow outside the boundary layer. This mainstream flow, accordingly, departs markedly from the potential-flow solution assumed for the boundary-layer calculations. Boundary-layer theory predicts the separation point accurately only in the case of streamlined bodies with relatively small wakes. Nevertheless, the circular cylinder is a good test case for checking the accuracy of boundary-layer computations.

Figure 9.1. Computed values of Λ and momentum thickness plotted against an angle around the cylinder's surface measured from the fore stagnation point.

Numerical solutions of the momentum-integral equation can also be found via Thwaites's method [79], which does not use the Pohlhausen approximate velocity profile as in Eq.(3.76) or Eq.(3.86). It is very simple to use and, for some applications, is more accurate than the Pohlhausen method. A suitable FORTRAN program for Thwaites's method is given by Cebeci and Bradshaw (1977) [80].

Computational methods based on the momentum-integral equation are available for the turbulent boundary layer. In this case, one or more semi-empirical relationships, in addition to the momentum-integral equation, are required. For example, most methods make use of the formula for $C_f$ due to Ludwieg and Tillmann (1949) [81]:

$C_f=0.246\times {10}^{-0.678H}{\left(\frac{U_e\theta }{v}\right)}^{-0.268}$

A good method of this type, due to Head (1958) [82], is relatively simple to use but performs better than many more complex methods based on the differential equations of motion. A FORTRAN program based on Head's method is also given by Cebeci and Bradshaw.

To begin computation of the turbulent boundary layer by Head's method, it is necessary to locate the transition point and supply initial values of θ and $H\equiv {\delta }^{⁎}/\theta$ . In Section3.6.7, we saw that, for the boundary layer on a flat plate, Eq.(3.154) holds at the transition point—that is, there is no discontinuous change in momentum thickness. This applies equally well to the more general case. Once the transition point is located, then the starting value for θ in the turbulent boundary layer is given by the final value in the laminar part. However, since transition is assumed to occur instantaneously at a specific location along the surface, there is a discontinuous change in velocity profile shape at the transition point, which implies a discontinuous change in the shape factor H. To a reasonable approximation,

$H_{\mathrm{Tt}}=H_{\mathrm{Lt}}-\mathrm{\Delta }H$

where

$\begin{array}{ccc}\hfill 2\mathrm{\Delta }H& =0.821+0.114{\log }_{10}({\mathit{Re}}_{\theta \mathrm{t}})\hfill & \hfill {\mathit{Re}}_{\theta \mathrm{t}}<5\times {10}^4\\ \hfill \mathrm{\Delta }H& =1.357\hfill & \hfill {\mathit{Re}}_{\theta \mathrm{t}}>5\times {10}^4\end{array}$

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Steady Separation of Incompressible Laminar Flow from Two-dimensional Surfaces

PAUL K. CHANG , in Separation of Flow, 1970

3 Discussion

For a prediction of the position of separation, von Karman's momentum integral equation is often used. This equation gives the approximate solution much more easily and quickly compared with exact solutions such as Görtler's because, by an integral across the boundary layer, the partial differential equation is reduced to an ordinary one. It is known that von Karman's momentum equation yields better results for accelerated flow than for decelerated flow, and the position of the separation point determined by the von Karman momentum equation is usually located downstream of the separation point computed by an exact solution.

In flow separation $\lambda =2\xi \frac{d}{d\xi }\ln u_e,$ where $\xi ={\displaystyle {\int }_0^x{u}_e\left(x\right)dx}$ is a very important value, as Meksyn [41] pointed out. These functions of λ and ξ are very closely connected with the principal function β(ξ₁) introduced by Görtler, because λ = u _e0β(ξ₁) and ${\xi }_1=\frac{1}{v}\xi$ .

Meksyn studied and discussed the condition of flow separation for Falkner–Skan's equation:

(31) ${\overline{f}}^{‴}+\overline{f}{\overline{f}}^{″}=\lambda \left(1-{\overline{f}}^2\right),$

where $\stackrel{=}{f}=\frac{\psi \left(x,y\right)\sqrt{Re}}{L{u}_e\left(x\right)g\left(x\right)}$ and g(x) is a dimensionless scale factor and refers to a partial derivative with respect to $\stackrel{=}{\eta }=\frac{y\sqrt{Re}}{L·g\left(x\right)}$ , where L is a reference length and λ is a constant to be determined.

If λ = 0, then this equation is reduced to Blasius' equation for flow over a flat plate. If λ = –1, then eqn. (31) represents two-dimensional stagnation flow, and if $\lambda =-\frac{2m}{m+1}$ , then eqn. (31) is for flow, u_e (x) = cx^m representing flow past a wedge in the neighborhood of the leading edge.

For a circular cylinder, in a potential flow solution based upon the inviscid flow velocity distribution of u_e /u _∞ = 2 sin φ, Hiemenz [42] evaluated λ _s = ≃1, but for real flow λ _s = 0·40 because, as Prandtl [43] found, the pressure around the separation point must satisfy certain conditions associated with the existence of reverse flow downstream of separation. Since λ is an important factor for separation, and there exists a difference of λ _s based upon potential and real flow, Hiemenz's solution based upon potential flow velocity may not be right, or its supposed correctness may be fortuitous. Hiemenz solved the separation along a circular cylinder by a power series and u_e (x) expressed by only three terms. Because by this expression the λ-curve does not have a maximum point, and λ increases monotonically for u_e (x), its derivatives du_e /dx and d ² u_e /dx² are not correct within the vital region of separation.

The correct expression of u_e (x) should be found not only by fitting the measured value of u_e (x), but it must also satisfy the two characteristic conditions that u_e (x) is maximum at φ ˜ 70° and u_e (x) is minimum, or at least has a horizontal tangent at φ ˜ 85°. These two conditions radically change the shape of du_e /dx and of λ close to separation. To satisfy these conditions, a large number of terms of the power series is needed to express u_e (x), and the three terms which Hiemenz used are not sufficient. However, the large number of terms of the power series makes the problem of separation intractable. Hiemenz's solution of separation, by using three terms only which gave the correct solution, is purely fortuitous. Meksyn [41] further states that the difficulty of using the power series for the separation is illustrated by a variation of f″(ξ, 0) with respect to x. If f″(ξ, 0) is plotted vs. x, then for a real flow close to the separation point the curve shown in Fig. 15 results.

FIG. 15. f″(ξ, 0) vs. x [41]

Such a curve cannot be expressed by a power series in x, because at x_s the tangent to the curve changes so rapidly that ∂/∂y ≫ ∂/∂x does not hold there. So in the immediate neighborhood of the point of separation, the main assumption of the boundary layer, ∂/∂y ≫ ∂/∂x, breaks down. Therefore the curve of f″(ξ, 0) vs. x is considered as consisting of two sections, one from the forward stagnation point to the separation point, and the other one starting from the separation point and downstream of it having a nearly constant f″(1,0). Then, f″(ξ, 0) is reduced to an algebraic equation and solved easily.

For the slender streamlined body, if the separation takes place close to the rear stagnation point, the pressure or velocity distribution of potential flow can be used as the approximation of viscous flow because the position of the maximum velocity and the separation point of real flow do not move much from those solutions obtained by potential flow.

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Boundary Layers and Related Topics

Pijush K. Kundu , ... David R. Dowling , in Fluid Mechanics (Sixth Edition), 2016

10.5 von Karman Momentum Integral Equation

Exact solutions of the boundary-layer equations are possible only in simple cases. In more complicated problems, approximate methods satisfy only an integral of the boundary-layer equations across the layer thickness. When this integration is performed, the resulting ordinary differential equation involves the boundary layer's displacement and momentum thicknesses, and its wall shear stress. This simple differential equation was derived by von Karman in 1921 and applied to several situations by Pohlhausen (1921).

The common emphasis of an integral formulation is to obtain critical information with minimum effort. The important results of boundary-layer calculations are the wall shear stress, displacement thickness, momentum thickness, and separation point (when one exists). The von Karman boundary-layer momentum integral equation explicitly links the first three of these, and can be used to estimate, or at least determine the existence of, the fourth. The starting points are (7.2) and (10.9), with the pressure gradient specified in terms of U _e (x) from (10.11) and the shear stress τ = μ(∂u/∂y):

(10.37) $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=U_e\frac{d{U}_e}{dx}+\frac{1}{\rho }\frac{\partial \tau }{\partial y}.$

Multiply (7.2) by u and add it to the left side of this equation:

(10.38) $u\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{\partial \left(u^2\right)}{\partial x}+\frac{\partial \left(vu\right)}{\partial y}=U_e\frac{d{U}_e}{dx}+\frac{1}{\rho }\frac{\partial \tau }{\partial y}.$

Move the term involving U _e to the other side of the last equality, and integrate (7.2) and (10.38) from y = 0 where u = v = 0 to y = ∞ where u = U _e and v = v _∞:

(10.39) $\underset{0}{\overset{\infty }{\int }}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)dy=0\to \underset{0}{\overset{\infty }{\int }}\frac{\partial u}{\partial x}dy=-\underset{0}{\overset{\infty }{\int }}\frac{\partial v}{\partial y}dy=-{\left[v\right]}_0^{\infty }=-v_{\infty },$

(10.40) $\underset{0}{\overset{\infty }{\int }}\left(\frac{\partial \left(u^2\right)}{\partial x}+\frac{\partial \left(vu\right)}{\partial y}-U_e\frac{d{U}_e}{dx}\right)dy=+\frac{1}{\rho }\underset{0}{\overset{\infty }{\int }}\frac{\partial \tau }{\partial y}dy\to \underset{0}{\overset{\infty }{\int }}\left(\frac{\partial \left(u^2\right)}{\partial x}-U_e\frac{d{U}_e}{dx}\right)dy+U_e{v}_{\infty }=-\frac{1}{\rho }{\tau }_w,$

where τ _w is the shear stress at y = 0 and τ = 0 at y = ∞. Use the final form of (10.39) to eliminate v _∞ from (10.40), and exchange the order of integration and differentiation in the first term of (10.40):

(10.41) $\frac{d}{dx}\underset{0}{\overset{\infty }{\int }}{u}^{2}dy-\underset{0}{\overset{\infty }{\int }}{U}_e\frac{d{U}_e}{dx}dy-U_e\underset{0}{\overset{\infty }{\int }}\frac{\partial u}{\partial x}dy=-\frac{1}{\rho }{\tau }_w.$

Now, note that

$U_e\underset{0}{\overset{\infty }{\int }}\frac{\partial u}{\partial x}dy=U_e\frac{d}{dx}\underset{0}{\overset{\infty }{\int }}udy=\frac{d}{dx}\left(U_e\underset{0}{\overset{\infty }{\int }}udy\right)-\frac{d{U}_e}{dx}\underset{0}{\overset{\infty }{\int }}udy.$

and use this to rewrite the third term on the left side of (10.41) to find:

(10.42) $\frac{d}{dx}\underset{0}{\overset{\infty }{\int }}\left(u^2-U_{e}u\right)dy+\frac{d{U}_e}{dx}\underset{0}{\overset{\infty }{\int }}\left(u-U_e\right)dy=-\frac{1}{\rho }{\tau }_w.$

A few final algebraic rearrangements produce:

(10.43) $\frac{1}{\rho }{\tau }_w=\frac{d}{dx}\left[U_e^2\underset{0}{\overset{\infty }{\int }}\frac{u}{U_e}\left(1-\frac{u}{U_e}\right)dy\right]+\frac{d{U}_e}{dx}{U}_e\underset{0}{\overset{\infty }{\int }}\left(1-\frac{u}{U_e}\right)dy,\text{or}\frac{1}{\rho }{\tau }_w=\frac{d}{dx}\left[U_e^2\theta \right]+U_e{\delta }^{\ast }\frac{d{U}_e}{dx}.$

Throughout these manipulations, U _e and dU _e /dx may be moved inside or taken outside the vertical-direction integrations because they only depend on x.

Equation (10.43) is known as the von Karman boundary-layer momentum integral equation, and it is valid for steady laminar boundary layers and for time-averaged flow in turbulent boundary layers. It is a single ordinary differential equation that relates three unknowns θ,δ∗, and τ _w , so additional assumptions must be made or correlations provided to obtain solutions for these parameters. The search for appropriate assumptions and empirical correlations was actively pursued by many researchers in the middle of the twentieth century starting with Pohlhausen (1921) and ending with Thwaites (1949) who combined analysis and inspired guesswork with the laminar boundary-layer measurements and equation solutions known at that time to develop the approximate empirical laminar-boundary-layer solution procedure for (10.43) described in the next section.

Example 10.5

Use the von Karman boundary-layer momentum integral equation to determine how the wall shear stress depends on downstream distance in an accelerating flow where U _e (x) = (U _o /L)x.

Solution

The given exterior flow velocity follows a power law with n = 1, so the generic Falkner-Skan boundary-layer thickness is:

$\delta (x)=\sqrt{\nu x/U_e\left(x\right)}=\sqrt{\nu L/U_o}=\mathit{const}.$

From this, we can deduce that θ and δ∗ are constant as well since are both defined as integrals of the velocity profile and are therefore proportional to the generic thickness, δ. For example:

$\theta \equiv \underset{y=0}{\overset{\infty }{\int }}\frac{u}{U_e}\left(1-\frac{u}{U_e}\right)dy=\delta \underset{y=0}{\overset{\infty }{\int }}{f}^{\prime }(\eta )\left(1-f^{\prime }(\eta )\right)d\eta =\delta ·\text{const}\text{.}$

where η and f(η) are defined by (10.34). Now use (10.43) and U _e (x) = (U _o /L)x to find:

$\frac{1}{\rho }{\tau }_w=\frac{d}{dx}\left[\frac{U_o^2{x}^2}{L^2}\theta \right]+\frac{U_{o}x}{L}{\delta }^{\ast }\frac{d}{dx}\left[\frac{U_{o}x}{L}\right]=\frac{2U_o^{2}x}{L^2}\theta +\frac{U_o^{2}x}{L^2}{\delta }^{\ast },$

and this can be mildly simplified to:

$\frac{{\tau }_w}{\frac{1}{2}\rho U_o^2}=\left(\frac{4\theta +2{\delta }^{\ast }}{L}\right)\frac{x}{L}.$

Thus, the skin friction increases linearly with downstream distance in this case. However, the three-unknowns-and-one-equation problem persists since values for θ and δ∗ are needed to determine τ _w . Thwaites method provides an approximate remedy for this problem.

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Viscous Flow and Boundary Layers

E.L. Houghton , ... Daniel T. Valentine , in Aerodynamics for Engineering Students (Seventh Edition), 2017

3.5.1 An Approximate Velocity Profile for the Laminar Boundary Layer

As explained in the previous subsection, an approximate expression is required for the velocity profile in order to use the momentum-integral equation. A reasonably accurate approximation can be obtained using a cubic polynomial in the form

(3.76) $\bar{u}(\equiv u/U_e)=a+b\bar{y}+c{\bar{y}}^2+d{\bar{y}}^3$

where $\bar{y}=y/\delta$ . To evaluate the coefficients a, b, c, and d, four conditions are required: two at $\bar{y}=0$ and two at $\bar{y}=1$ . Of these conditions, two are readily available:

(3.77) $\bar{u}=0\text{at}\bar{y}=0$

(3.78) $\bar{u}=1\text{at}\bar{y}=1$

In real boundary-layer velocity profiles (see Fig. 3.6), velocity varies smoothly to reach $U_e$ ; there is no kink at the edge of the boundary layer. It follows then that the velocity gradient is zero at $y=\delta$ , giving a third condition:

(3.79) $\frac{\partial \bar{u}}{\partial \bar{y}}=0\text{at}\bar{y}=1$

To obtain the fourth and final condition, it is necessary to return to the boundary-layer Eq.(3.10). At the wall $y=0$ , $u=v=0$ , so both terms on the left-hand side are zero at $y=0$ . Noting that $\tau =\mu \partial u/\partial y$ , the required condition is thus given by

(3.80) $\frac{dp}{dx}=\frac{\partial \tau }{\partial \mathrm{y}}\text{at}y=0$

Since $y=\bar{y}\delta$ and $p+\rho U_e^2=\text{const.}$ , this equation can be rearranged to read

(3.81) $\frac{{\partial }^2\bar{\nu }}{\partial {\bar{y}}^2}=-\frac{{\delta }^2}{v}\frac{d{U}_e}{dx}\text{at}\bar{y}=0$

In terms of coefficients a, b, c, and d, the four conditions Eq.(3.77), Eq.(3.78), Eq.(3.79), and Eq.(3.81) become

(3.82) $a=0$

(3.83) $b+c+d=1$

(3.84) $b+2c+3d=0$

(3.85) $2c=-\mathrm{\Lambda }\text{where}\mathrm{\Lambda }\equiv \frac{{\delta }^2}{\nu }\frac{d{U}_e}{dx}$

Equations(3.83) to(3.85) are easily solved for b, c, and d to give the following approximate velocity profile:

(3.86) $\bar{u}=\frac{3}{2}\bar{y}-\frac{1}{2}{\bar{y}}^3+\frac{\mathrm{\Lambda }}{4}(\bar{y}-2{\bar{y}}^2+{\bar{y}}^3)$

In Eq.(3.86), Λ is often called the Pohlhausen parameter. It determines the effect of an external pressure gradient on the shape of the velocity profile. $\mathrm{\Lambda }>0$ and <0 correspond, respectively, to favorable and unfavorable pressure gradients. For $\mathrm{\Lambda }=-6$ , the wall shear stress ${\tau }_{\mathrm{w}}=0$ ; for more negative values of Λ, flow reversal at the wall develops. Thus $\mathrm{\Lambda }=-6$ corresponds to boundary-layer separation. Velocity profiles corresponding to various values of Λ are plotted in Fig. 3.20, in which the flat-plate profile corresponds to $\mathrm{\Lambda }=0$ ; $\mathrm{\Lambda }=6$ for the favorable pressure gradient; $\mathrm{\Lambda }=-4$ for the mild adverse pressure gradient; $\mathrm{\Lambda }=-6$ for the strong adverse pressure gradient; and $\mathrm{\Lambda }=-9$ for the reversed-flow profile.

Figure 3.20. Laminar velocity profile.

For the flat-plate case $\mathrm{\Lambda }=0$ , we compare the approximate velocity profile of Eq.(3.86) with two other approximate profiles in Fig. 3.20. The profile labeled Blasius is the accurate solution of the differential equations of motion given in Section 3.2.1 and in Fig. 3.6.

The various quantities introduced in Sections 3.2.2 and3.2.3 can be evaluated using the approximate velocity profile Eq.(3.86). For example, if Eq.(3.86) is substituted in turn into Eqs.(3.24), (3.26), and(3.31), the following are obtained using Eq.(3.30):

(3.87) $\begin{array}{ccc}\hfill I_1=\frac{{\delta }^{⁎}}{\delta }& \hfill =\hfill & \underset{0}{\overset{1}{\int }}[1-(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})\bar{y}+\frac{\mathrm{\Lambda }}{2}{\bar{y}}^2+(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4}){\bar{y}}^3]d\bar{y}\hfill \\ \hfill & \hfill =\hfill & {[\bar{y}-(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})\frac{{\bar{y}}^2}{2}+\frac{\mathrm{\Lambda }}{2}\frac{{\bar{y}}^3}{3}+(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4})\frac{{\bar{y}}^4}{4}]}_0^1\hfill \\ \hfill & \hfill =\hfill & 1-(\frac{3}{4}+\frac{\mathrm{\Lambda }}{8})+\frac{\mathrm{\Lambda }}{6}+(\frac{1}{8}-\frac{\mathrm{\Lambda }}{16})\hfill \\ \hfill & \hfill =\hfill & \frac{3}{8}-\frac{\mathrm{\Lambda }}{48}\hfill \end{array}$

(3.88) $\begin{array}{ccc}\hfill I=\frac{\theta }{\delta }& \hfill =\hfill & \underset{0}{\overset{1}{\int }}[(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})\bar{y}-\frac{\mathrm{\Lambda }}{2}{\bar{y}}^2-(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4}){\bar{y}}^3]\hfill \\ \hfill & \hfill \hfill & \times [1-(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})\bar{y}+\frac{\mathrm{\Lambda }}{2}{\bar{y}}^2+(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4}){\bar{y}}^3]d\bar{y}\hfill \end{array}$

(3.89) $\begin{array}{ccc}\hfill & \hfill =\hfill & [(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})\frac{{\bar{y}}^2}{2}-\{{(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})}^2+\frac{\mathrm{\Lambda }}{2}\}\frac{{\bar{y}}^3}{3}\hfill \\ \hfill & \hfill \hfill & +\{2(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})\frac{\mathrm{\Lambda }}{2}-(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4})\}\frac{{\bar{y}}^4}{4}\hfill \\ \hfill & \hfill \hfill & +\{2(\frac{3}{2}+\frac{\mathrm{\Lambda }}{4})(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4})-\frac{{\mathrm{\Lambda }}^2}{4}\}\frac{{\bar{y}}^5}{5}\hfill \\ \hfill & \hfill \hfill & {-\left\{2\frac{\mathrm{\Lambda }}{2}(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4})\right\}\frac{{\bar{y}}^6}{6}-{(\frac{1}{2}-\frac{\mathrm{\Lambda }}{4})}^2\frac{{\bar{y}}^7}{7}]}_0^1\hfill \\ \hfill & \hfill =\hfill & \frac{1}{280}(39-\frac{\mathrm{\Lambda }}{2}-\frac{1}{6}{\mathrm{\Lambda }}^2)\hfill \end{array}$

(3.90) $C_f=\frac{{\tau }_{\mathrm{w}}}{\frac{1}{2}\rho U_e^2}=\frac{2\mu }{\rho U_e^2}{\left(\frac{\partial u}{\partial y}\right)}_{\mathrm{w}}=\frac{2\mu }{\rho U_e\delta }{\left(\frac{d\bar{u}}{d\bar{y}}\right)}_{\bar{y}=0}$

(3.91) $=\frac{\mu }{\rho U_e\delta }(3+\frac{\mathrm{\Lambda }}{2})$

Quantities $I_1$ and I depend only on the shape of the velocity profile; for this reason, they are usually known as shape parameters. If the more accurate differential form of the boundary-layer equations were used, rather than the momentum-integral equation with approximate velocity profiles, the boundary-layer thickness δ would become rather less precise. For this reason, it is more common to use the shape parameter $H={\delta }^{⁎}/\theta$ . H is frequently referred to simply as the shape parameter.

For the numerical methods discussed in Section 9.1, which are used in the general case with an external pressure gradient, it is preferable to employ a somewhat more accurate quartic polynomial as the approximate velocity profile, particularly for predicting the transition point. The quartic velocity profile is derived in a very similar way to that given earlier, with the main differences being the addition of another term $e{\bar{y}}^4$ on the right-hand side of Eq.(3.76) and the need for an additional condition at the edge of the boundary layer. This latter difference requires that

(3.92) $\frac{d^2\bar{u}}{d{\bar{y}}^2}=0\text{at}\bar{y}=1$

which has the effect of making the velocity profile even smoother at the edge of the boundary layer, thereby improving the approximation. The resulting quartic velocity profile takes the form

(3.93) $\bar{u}=2\bar{y}-2{\bar{y}}^3+{\bar{y}}^4+\frac{\mathrm{\Lambda }}{6}(\bar{y}-3{\bar{y}}^2+3{\bar{y}}^3-{\bar{y}}^4)$

Using this profile and following procedures similar to those outlined previously leads to the following expressions:

(3.94) $I_1=\frac{3}{10}-\frac{\mathrm{\Lambda }}{120}$

(3.95) $I=\frac{1}{63}(\frac{37}{5}-\frac{\mathrm{\Lambda }}{15}-\frac{{\mathrm{\Lambda }}^2}{144})$

(3.96) $C_f=\frac{\mu }{\rho U_e\delta }(4+\frac{\mathrm{\Lambda }}{3})$

Note that it follows from(3.96) that, with the quartic velocity profile, the separation point where ${\tau }_{\mathrm{w}}=0$ now corresponds to $\mathrm{\Lambda }=-12$ .

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Governing differential equations for natural circulation systems

Pallippattu Krishnan Vijayan , ... Naveen Kumar , in Single-Phase, Two-Phase and Supercritical Natural Circulation Systems, 2019

3.3.1 Closed-loop single-phase natural circulation systems

Fig. 3.1 represents the geometry and coordinate system used for the single-phase NCS under consideration. In general, single-phase liquids are considered incompressible, and therefore the mass conservation equation can be rewritten as

(3.14) $\frac{\partial u}{\partial s}=0$

The above equation implies that the velocity is independent of position in a uniform-diameter loop and is only a function of time. Often, it is convenient to work with the integral momentum equation. The integral momentum equation can be obtained by integrating Eq. (3.7) over a closed loop to yield the following equation:

(3.15) $L_t\rho \frac{du}{dt}+\frac{f{L}_t\rho u^2}{2D}+g\oint \rho dz=0$

Note that $ds\sin \theta =dz$ has been used in the above equation. As the sum of the pressure drop over a closed loop is zero, the pressure term vanishes from the momentum equation, which is the main advantage of the integral momentum equation as would be seen later while numerically solving it. Furthermore, the sum of the acceleration pressure drop also becomes zero over the closed loop. Additionally, if Boussinesq approximation is valid, then the fluid properties are assumed constant except for the density in the buoyancy force term which can be expressed as

(3.16a) $\rho ={\rho }_0\left[1-{\beta }_T\left(T-T_0\right)\right]$

where

(3.16b) ${\beta }_T=\frac{1}{v}{\left(\frac{\partial v}{\partial T}\right)}_p$

Note that ρ ₀ is a reference density at the reference temperature T ₀. It may be mentioned that the Boussinesq approximation is valid for only small temperature differences. Use of Boussinesq approximation was found to give a good match with test data for temperature differences between the hot and cold legs as high as 30°C if the loop average temperature was used as the reference value. Using this, Eq. (3.15) becomes

(3.17) $L_t{\rho }_0\frac{du}{dt}+\frac{f{L}_t{\rho }_0{u}^2}{2D}-{\rho }_0{\beta }_{T}g\oint Tdz=0$

Note that Eq. (3.17) is an ordinary differential equation. Often, the friction coefficient can be expressed as

(3.18) $f=p/\mathit{R}{\mathit{e}}^b$

Using this, the momentum equation can be rewritten as

(3.19) $L_t{\rho }_0\frac{du}{dt}+\frac{p{L}_t{\rho }_0^{1-b}{\mu }^b{u}^{2-b}}{2D^{1+b}}-{\rho }_0{\beta }_{T}g\oint Tdz=0$

For constant fluid properties, Eq. (3.13) can be rewritten as

(3.20) $\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial s}-{\alpha }_d\frac{{\partial }^{2}T}{\partial s^2}=\left\{\begin{array}{l}\frac{4q_h}{D{\rho }_{0}Cp}\mathrm{heater}\mathrm{for}0<s\le s_h\\ 0\mathrm{pipes}\mathrm{for}{s}_h<s\le s_{hl}\mathrm{and}{s}_c<s\le s_t\\ \frac{-4U_i}{D{\rho }_{0}Cp}\mathrm{cooler}\mathrm{for}{s}_{hl}<s\le s_c\end{array}\right\}$

Note that the fluid properties in the hot leg, cold leg, the source and the sink are all different. Hence it is important to select an appropriate reference temperature to calculate the reference density. Since the friction pressure drop occurs over the entire loop as well as the driving buoyancy pressure differential depends on the temperature integral over the entire loop, it is appropriate to use the loop average temperature as the reference value to calculate the fluid properties.

Since the heat transport capability of natural circulation loops is directly proportional to the generated mass flow rate, w, often the governing equations are expressed in terms of the mass flow rate. Therefore the conservation equations of mass, momentum, and energy can be rewritten as

(3.21) $\frac{\partial w}{\partial s}=0$

(3.22) $\frac{L_t}{A}\frac{dw}{dt}+\frac{p{L}_t{\mu }^b{w}^{2-b}}{2D^{1+b}{\rho }_0{A}^{2-b}}-{\rho }_0{\beta }_{T}g\oint Tdz=0$

(3.23) $\frac{\partial T}{\partial t}+\frac{w}{A{\rho }_0}\frac{\partial T}{\partial s}-{\alpha }_d\frac{{\partial }^{2}T}{\partial s^2}=\left\{\begin{array}{l}\frac{4q_h}{D{\rho }_{0}Cp}\mathrm{heater}\mathrm{for}0<s\le s_h\\ 0\mathrm{pipes}\mathrm{for}{s}_h<s\le s_{hl}\mathrm{and}{s}_c<s\le s_t\\ -\frac{4U_i\left(T-T_s\right)}{D{\rho }_{0}Cp}\mathrm{cooler}\mathrm{for}{s}_{hl}<s\le s_c\end{array}\right\}$

For ordinary fluids like water, the axial conduction term can be neglected from Eqs. (3.20) and (3.23). In addition, the pipes are considered to be adiabatic in the above derivation. Often, some natural convective heat losses are unavoidable through the insulated pipes. Accounting for this, the energy equation can be rewritten as

(3.24) $\frac{\partial T}{\partial t}+\frac{w}{A{\rho }_0}\frac{\partial T}{\partial s}=\left\{\begin{array}{l}\frac{4q_h}{D{\rho }_{0}Cp}\mathrm{heater}\mathrm{for}0<s\le s_h\\ -\frac{4U_a}{D{\rho }_{0}Cp}\mathrm{pipes}\mathrm{for}{s}_h<s\le s_{hl}\mathrm{and}{s}_c<s\le s_t\\ -\frac{4U_i\left(T-T_s\right)}{D{\rho }_{0}Cp}\mathrm{cooler}\mathrm{for}{s}_{hl}<s\le s_c\end{array}\right\}$

where U _a is the overall heat transfer coefficient to account for the ambient heat losses through the insulated pipe. If the closed loop NCS is formed by two heat exchangers (HXs), then the energy equation is modified as (see also Fig. 3.5),

Figure 3.5. NCL with source and sink formed by HXs.

(3.25) $\frac{\partial T}{\partial t}+\frac{w}{A{\rho }_0}\frac{\partial T}{\partial s}=\left\{\begin{array}{l}\frac{4U_h\left(T-T_h\right)}{D{\rho }_{0}Cp}\mathrm{heater}\mathrm{for}0<s\le s_h\\ -\frac{4U_a\left(T-T_a\right)}{D{\rho }_{0}Cp}\mathrm{pipes}\mathrm{for}{s}_h<s\le s_{hl}\mathrm{and}{s}_c<s\le s_t\\ -\frac{4U_i\left(T-T_c\right)}{D{\rho }_{0}Cp}\mathrm{cooler}\mathrm{for}{s}_{hl}<s\le s_c\end{array}\right\}$

where U _h is the overall heat transfer coefficient based on the inside heat transfer area of the heater.

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