Understanding the Physics of the Ideal Form of Fluid Dynamics (Perfect Fluid) -7-
Regarding the perfect fluid, which is considered one of the ideal states in fluid dynamics. When assuming a continuum, a perfect fluid is defined as one where the tangential stress (resistance) of the fluid is ignored.
When fluid pressure is expressed as a second-order tensor, it is represented as follows, combined with the scalar quantity of pressure (p) and the Kronecker delta (a process that assigns values to the diagonal components of a matrix).
$${p_{ij}=-p\delta_{ij}}$$
In this series, we will focus on the physical properties of a perfect fluid.
Last time, we looked at the flow from the physics (analysis) of potential flow to determining mechanical problems (such as generated forces).
Starting from this time, we will look at two-dimensional problems of potential flow. Since we are focusing on two-dimensional problems, we will introduce a discussion using complex velocity potential.
Complex Velocity Potential
We limit ourselves to a two-dimensional plane (x-y plane) relative to a Cartesian coordinate system. Since the flow field also becomes a planar problem, the vorticity is as follows.
$${\bm{\omega}=\textrm{rot}{\bm{u}}=(0,0,\omega)}$$ , $${\bm{u}=(u,v,0)}$$
In other words, the vorticity (vector) is always perpendicular to the flow field. The magnitude of the vorticity, when using a stream function, is as follows.
$${\omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}=-(\frac{\partial^2 \Psi}{\partial x^2}+\frac{\partial^2 \Psi}{\partial y^2})=-\Delta{\Psi}}$$
Since potential flow is premised on being an incompressible perfect fluid with no vorticity, the result of the Laplacian for the stream function becomes zero.
From here, we will consider the problem by incorporating complex functions. Ultimately, for two-dimensional potential flow problems, the relationship between the velocity potential and the stream function corresponds to the Cauchy-Riemann equations shown below.
$${u=\frac{\partial \Phi}{\partial x}=\frac{\partial \Psi}{\partial y}}$$, $${v=\frac{\partial \Phi}{\partial y}=-\frac{\partial \Psi}{\partial x}}$$
From here, we introduce the complex velocity potential in a form that includes complex functions.
$${W=\Phi+{i}\Psi}$$, $${z=x+{i}y}$$
Assuming it is a holomorphic function, the differential form of the complex velocity potential is as follows.
$${\frac{dW}{dz}=\frac{\partial W}{\partial x}=\frac{\partial \Phi}{\partial x}+{i}\frac{\partial \Psi}{\partial x}=u-{i}v}$$
When we re-introduce complex velocity for the complex velocity potential, the relationship between the two is as follows.
$${w=u-{i}v=\frac{dW}{dz}}$$
As described above, if the complex velocity potential is given as a holomorphic function, it can be said that the explanation of two-dimensional potential flow is possible. In that case, it follows the Cauchy-Riemann relations shown earlier.
Analysis of simple complex velocity potentials
When there are multiple holomorphic complex velocity potentials, their linear combination (where coefficient C is an arbitrary constant) can also be said to be a holomorphic function.
$${W=C_1W_1+C_2W_2}$$
When a simple two-dimensional irrotational flow is found, a complex solution can be constructed by superimposing multiple complex velocity potentials. Here, we introduce several practical examples.
Regarding the simplest uniform flow (treating the speed of the flow field U as a constant), when an inclination with the x-axis is provided, the complex velocity potential is expressed as follows.
$${W=Ue^{-{i}\alpha}z}$$
Next, regarding flow with a corner, we define the complex velocity potential using a power function as follows.
$${W=Az^a}$$, $${z=re^{{i}\theta}, A=|A|e^{{i}\alpha}}$$
From the relationship between this complex velocity potential, the velocity potential, and the stream function (note that these are not complex functions), it becomes as follows.
$${\Phi=|A|r^a\textrm{cos}(a\theta+\alpha)}$$, $${\Psi=|A|r^a\textrm{sin}(a\theta+\alpha)}$$
Finding the angle to make the stream function zero from here results in the following. The streamlines are a group of radial straight lines passing through the origin, based on the angle ($${\pi/a}$$) between two adjacent straight lines representing the corner.
$${\theta=\frac{n\pi-\alpha}{a}}$$
Other streamlines are obtained by setting the stream function to a constant value (C), as follows.
$${r=|\frac{C}{A}|^{\frac{1}{a}}|\textrm{sin}(a\theta+\alpha)|^{-\frac{1}{a}}}$$
The representation of the corner is determined by the positive constant (a). For example, if the positive constant (a) is a value greater than 1, the corner is naturally defined as an acute angle.
Finally, regarding a two-dimensional doublet (a two-dimensional source and sink of equal strength placed at a close distance), taking the limit with respect to the close distance, the complex velocity potential becomes as follows.
$${W=-\frac{\mu{e^{{i}\alpha}}}{z}}$$, $${z_1=\varepsilon{e^{{i}\alpha}}, z_2=-\varepsilon{e^{{i}\alpha}} }$$
Here, assuming the strength (m) of the source (sink), we define the distance ($${\varepsilon}$$) from the origin.
$${\mu=2\varepsilon{m}}$$
In a two-dimensional flow caused by a doublet, all circles tangent to the axis at the origin become streamlines, and all circles orthogonal to the axis at the origin become equipotential lines.
Conclusion
In this installment, we considered the derivation of simple flow fields for two-dimensional potential flow problems based on the concept of complex functions (Cauchy-Riemann equations).
Next time, we will deal with the problem of two-dimensional potential flow when a cylinder is used as an obstacle. I believe it will follow the flow based on the process covered this time.
-------------------------
Thank you for reading until the end. I hope this article has provided you with new insights for your life. I look forward to your continued support.
-------------------------
⭐︎⭐︎⭐︎ Shin Taniguchi's Profile ⭐︎⭐︎⭐︎
⭐︎⭐︎⭐︎ Roadmap ⭐︎⭐︎⭐︎