Approximate Analytic Solution of a Potential Flow Around an Obstacle

It is commonly acknowledged that jet type free surface flows are widely encountered in such diverse industrial and urban applications as engine combustion chambers, jet pumps, as well as reservoirs and dams. This type of flow has been the subject of a large number of theoretical, experimental and numerical studies. For the sake of example, one may cite the flow of a liquid emerging from an orifice of a tank. This kind of problems is generally difficult to solve explicitly due to the nonlinear condition that is imposed on the free boundary of an unknown shape. It was shown that the difficulty in solving such a problem increases with the complexity of the geometry of the flow domain. Its solution also depends on the properties of the fluid and on the flow conditions by Gasmi [1].

During the nineteenth century, the theory of complex variables made it possible to theoretically investigate the two-dimensional free surface flow, provided that the flow domain is polygonal and the fluid is irrotational, incompressible and non-viscous, and the effects of gravity are neglected. These hypotheses indeed allowed using the complex potential theory to solve two-dimensional flow problems within polygonal domains.

Over the past forty years, great attention has been devoted to studying the effect of the obstacle’s shape on the two-dimensional free surface flow of fluids. Bernoulli’s equation is generally utilized to describe some. Nonlinear problems in which the shape of the free surface is unknown

Furthermore, it was shown that various analytical and numerical methods can be used to determine the shape of the free surface flow for many problems dealing with the potential flow around different obstacle shapes by applying various techniques that are based, for example, on the conformal mapping transformation, the truncation series, and the boundary integral methods. It has indeed been reported that these techniques have been adopted by many authors, such as Forbes and Schwartz [2] and Birkhoff and Zarantoello [3], in order to determine the nonlinear solutions of subcritical and supercritical flows around a semi-circular obstacle. In this context, Gasmi and Mekias [4], Gasmi and Amara [5], Benjamin [6], and Vanden-Broeck [7, 8] considered the problems of flow within a channel with cavities. As for Sekhri et al. [9], they carried out some numerical studies to find the solution of two-dimensional potential flow problems with an immersed triangular obstacle. With regard to Merzougui and Laiadi [10], they investigated the problem of a fluid flowing over a triangular depression. Similarly, Dias and Vanden-Broeck [11] as well as Vanden-Broeck and Killer [12] succeeded in identifying the subcritical and supercritical free-surface solution of flows past a triangular obstacle. Likewise, Hanna et al. [13] adopted a numerical method that is based on a series truncation technique for a super-critical case in order to successfully solve the problem of a super-critical flow over a trapezoidal obstacle. On the other hand, many researchers [7-18] established a method that is based on the Shwartz-Christoffel transformation to solve similar problems. Furthermore, Smith and Lim [19] considered the supercritical flow around an obstacle that has a polygonal symmetrical shape and placed on the bottom of a channel. The conformal mapping and Hilbert’s method of a mixed boundary value problem were applied in the upper half-plane in order to determine the nonlinear surface profiles.

The present work focuses on a two-dimensional steady, irrotational and inviscid potential flow within a channel that has an obstacle placed on its bottom. The obstacle has a trapezoidal shape and makes an angle $\beta=\frac{\pi}{3}$ with the horizontal plane. It is important to note that, far upstream, the fluid is uniform, with a constant velocity U and uniform depth L. The free streamline theory, which is based on the Shwartz-Christoffel transformation and Kirchhoff theory, is then utilized to calculate the shape of the free surface flow of the fluid past a submerged obstacle. The findings indicate that the method used for that purpose can be easily implemented, and is highly effective in providing analytical solutions for these kinds of problems.

The above mentioned method has been successfully utilized by Vanden-Broeck [7, 8], Gasmi [1], Peng and Parker [20], and other researchers to determine the analytical solutions of free surface flow problems.

The problem at hand is formulated in Section 2, while Section 3 describes the method used to solve that problem. As for Section 4, it presents and discusses the results obtained. Finally, some concluding remarks about the problem are drawn, and future research suggestions are presented in Section 5.

The free streamline theory, which was initially introduced by Kirchhoff and was based on the hodograph mapping and Schwartz-Christoffel transformation, was utilized. It is noteworthy that this approach has already been adopted by Batchelor [22], Birkhoff and Zarantoello [3], Sekhri et al. [9] and other researchers to determine the exact solutions of problems in the case where the effects of the superficial tension and gravity force are neglected.

Therefore, the complex transformation may be expressed as:

$\Gamma=\log \left(\frac{U d z}{d f}\right)=\log \left(\frac{U}{u-i v}\right)=\log \left(\frac{U}{q}\right)+i \theta$    (5)

here, z=x+iy, q and θ are the fluid speed and the angle between the velocity vector and the horizontal plane, respectively. Consequently, the field occupied by the fluid in the plane Z is transformed into a semi-band in the plane Γ, as shown in Figure 4.

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Figure 4. The plane Γ

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Figure 5. The plane λ

Moreover, the Schwartz-Christoffel transformation method may be used to transform the half-band plane, as shown in Figure 3, into a lower half-plane that is illustrated in Figure 5.

The hodograph method and Schwartz-Christoffel mapping are used to write the relation below:

$\frac{\mathrm{d} \Gamma}{d \lambda}=A \prod_i\left(\lambda-\lambda_i\right)^{\frac{\alpha_i}{\pi}-1}$

where, λ_i is the coordinate of the plane λ- with respect to the vertex of the polygon, α_i is the corresponding internal angle in the plane Γ, and A is a constant. The previous transformation allows mapping the fluid region in the plane Γ into the upper half λ plane and the fluid region onto the real axis of the λ plane.

Considering this transformation, and in order to achieve a unique mapping, it was decided to take, for example, the points:

A: $\Gamma=0 \rightarrow \lambda=0$

$S: \Gamma=-\frac{i \pi}{3}=-i \beta \rightarrow \lambda=1$

The following expression may also be considered:

$\frac{\mathrm{d} \Gamma}{d \lambda}=\lambda^{\frac{-1}{2}}(\lambda-1)^{\frac{-1}{2}} \Rightarrow \int \mathrm{d} \Gamma=\int \frac{d \lambda}{\lambda^{\frac{1}{2}}(\lambda-1)^{\frac{1}{2}}}$

In order to calculate the above integral, the new variable $t=\sqrt{1-\lambda}$ in then introduced. Therefore:

$\lambda-1=t^2, \quad d \lambda=-2 t d t$

Regarding the transformation Γ, it can be written in the form:

$\Gamma=\frac{-2 M}{i} \int \frac{d t}{\sqrt{1-t^2}}$

Some simple mathematical operations may then be carried out to get:

$\Gamma=\frac{-2 M}{i} \times \arcsin (\sqrt{t})+N$

Which gives:

$\Gamma=\frac{-2 M}{i} \times \arcsin (\sqrt{1-\lambda})+N$    (6)

where, $N$ and $M$ are two constants to be determined.

For: $\lambda=1 \rightarrow \Gamma=-\frac{i \pi}{3}, N=-\frac{i \pi}{3}$

For: $\lambda=0 \rightarrow \Gamma=0, M=\frac{1}{3}$

Eq. (6) may therefore be written as:

$\Gamma=\frac{2 i}{3} \times \arcsin (\sqrt{1-\lambda})-\frac{i \pi}{3}$    (7)

Eq. (7) can be expressed as:

$\Gamma=\frac{2}{3} \ln (\sqrt{\lambda}+i \sqrt{1-\lambda})-\frac{i \pi}{3}$    (8)

For the purpose of identifying the transformation that can be used to transform the interior of infinite bands in the plane λ, as shown in Figure 5, into the lower half of the λ plane, the Schwartz-Christoffel transformation is used again, while selecting the same points A and S.

The following equation is then obtained:

$\frac{d f}{d \lambda}=\alpha \lambda^{-1}(\lambda-1)^{-1} \Rightarrow f=\alpha \int \frac{d \lambda}{\lambda(\lambda-1)}$

Then, after some mathematical operations, the relation between λ and f is then found as:

$f=\alpha \ln \left(\frac{\lambda-1}{\lambda}\right)+b$

where, α and b are two constants which can be calculated as follow:

$\cdot \lim _{\lambda \rightarrow+\infty} f=0$. Then: $b=0$

$\cdot \lim _{\lambda \rightarrow 0} \lambda\left(\alpha \frac{d f}{d \lambda}\right)=\frac{1}{3}$. Then: $\alpha=-\frac{L U}{\pi}.$

$f=-\frac{L U}{\pi} \ln \left(\frac{\lambda-1}{\lambda}\right)$    (9)

Afterwards, the relations given below are used:

$U \frac{d z}{d \lambda}=U \frac{d z}{d f} \frac{d f}{d \lambda}$    (10)

$\frac{d f}{d \lambda}=\frac{-L U}{\pi \lambda(\lambda-1)}$    (11)

On the other hand, transformation Γ may be expressed as:

$\Gamma=\log \left(\frac{U d z}{d f}\right)$    (12)

Therefore:

$\begin{gathered}U \frac{d z}{d \lambda}=\exp (\Gamma)=\exp \left(\frac{2}{3} \ln (\sqrt{\lambda}+i \sqrt{1-\lambda})-\frac{i \pi}{3}\right)= \\ \exp \left(-\frac{i \pi}{3}\right) \times(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2}{3}}\end{gathered}$

Next, using relations (10), (11) and (12) allows writing:

$\frac{d z}{d \lambda}=\frac{L U \exp \left(\frac{-i \pi}{3}\right)}{\pi} \frac{(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2}{3}}}{\lambda(\lambda-1)}$    (13)

Then, the relation (13) is employed to write the following integral:

$\int d z=\frac{L U \exp \left(\frac{-i \pi}{3}\right)}{\pi} \int \frac{1}{\lambda(\lambda-1)}(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2}{3}} d \lambda$    (14)

Which gives:

$z=\frac{\operatorname{LUexp}\left(\frac{-i \pi}{3}\right)}{\pi} \int \frac{1}{\lambda(\lambda-1)}(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2}{3}} d \lambda$    (15)

The free surface may therefore be presented parametrically as:

$\left\{\begin{array}{l}x={real}\left(\frac{L U \exp \left(\frac{-i \pi}{3}\right)}{\pi} \int \frac{1}{\lambda(\lambda-1)}(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2}{3}} d \lambda\right) \\ y={imag}\left(\frac{L U \exp \left(\frac{-i \pi}{3}\right)}{\pi} \int \frac{1}{\lambda(\lambda-1)}(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2}{3}} d \lambda\right)\end{array}\right.$

Performing the same calculations, for any angle β such that 0<β≤π, the shape of the free surface can generally be expressed in the following form:

$z=\frac{L U \exp (-i \beta)}{\pi} \int \frac{1}{\lambda(\lambda-1)}(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2 \beta}{\pi}} d \lambda$    (16)

which can be expressed parametrically as:

$\left\{\begin{array}{l}x={real}\left(\frac{L U \exp (-i \beta)}{\pi} \int \frac{1}{\lambda(\lambda-1)}(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2 \beta}{\pi}} d \lambda\right) \\ y={imag}\left(\frac{L U \exp (-i \beta)}{\pi} \int \frac{1}{\lambda(\lambda-1)}(\sqrt{\lambda}+i \sqrt{1-\lambda})^{\frac{2 \beta}{\pi}} d \lambda\right)\end{array}\right.$

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Figure 6. Shape of the free surface for beta=pi/2 ($\beta=\frac{\pi}{2}$)

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Figure 7. Shape of the free surface for beta=pi/4 ($\beta=\frac{\pi}{4}$)

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Figure 8. Shape of the free surface for beta=pi/6 ($\beta=\frac{\pi}{6}$)

For the sake of example, the graphical representations of the shape of the free surface flow, for some values of the inclination angle β, are given as Figures 6-8.

Until now, researchers have made a lot of effort to prove the existence and unity of the exact solution. Note that the simultaneous presence of a nonlinear term (a non-local term) and a diffusion term makes the Navier and Stokes equations more complicated. They therefore resorted to solving nonlinear equations. Once the solution was found, they then neglected the nonlinear term in order to find the exact solution.

The streamline method was applied in the present work in view of identifying the exact solution to the problem of finding the shape of the free surface flow. The adopted method is based on the hodograph method and the Schwartz-Christoffel transformation technique. The transformations utilized here are considered as fundamental approaches in fluid mechanics, as they make studies of difficult problems in the field of fluids mechanic simpler and easier to solve. They can be used to successfully transform complex flow domains into simpler ones. They can also help to simplify all calculations in order to obtain the exact solution to the problem of finding the shape of a free surface flow. It should be highlighted that the gravity and superficial tension are neglected in this work. These two conditions first made the calculations simpler as all non-linear terms disappeared from the equations used, and second, they contributed to finding the exact solution easily.

This study adopted the free streamline theory, which is based on the Kirchhoff and Hodograph transformations, for the purpose of finding an approximate analytical solution for different values of inclination angle β when a symmetric trapezoidal obstacle is used.

In our opinion, and according to the results obtained in this work, it can be concluded that the calculation approach used in this paper is highly efficient as it provides adequate solutions to this type of free surface flow problems. Moreover, this method may also be utilized to compare the exact solutions to the approximate solutions found in the case where gravity is taken into account, when the Froude number tends towards ∞, and/or in the case where the superficial tension is taken into consideration, when the Weber number tends towards ∞.

In a forthcoming work on this topic, we intend to present and study related problems dealing with the field of potential flows over different forms of obstacles, while considering new boundary conditions and adopting other numerical methods, such as the finite-volume method, in order to find approximate solutions and shapes of free surface flows.