Compuational Fluid Dynamics for Incompressible Flows

Week-02 (Live Session)

Durga Prasad Pydi

2026-01-31

Accessing Live Session Materials

Live Session timings: 05:00 PM - 07:00 PM IST on every Saturday Meeting Link: Will be circulated to enrolled students on their registered email id

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Course Contents

~~Applications of CFD and brief recap of governing equations and boundary conditions~~
Classification of PDEs - elliptic, parabolic and hyperbolic - highlighting the relevance to CFD
Key qualifications for solvers and Finite difference method (FDM) as a tool to solve PDEs
Application of FDM
- elliptic
- parabolic
- hyperbolic
Evaluating the learned strategies using techniques presented in 3
Alternate formulations of governing equation and their solutions
MAC Algorithm
Finite Volume Method using SIMPLE (last 4-5 weeks)

Concepts

Classification of Phenomena

For a general $2^{nd}$ order PDE, defined as $a \phi_{x x} + b \phi_{x y} + c \phi_{yy}=h$

The characteristics are given by $\frac{dy}{dx}=\frac{b \pm \sqrt{ b^{2}-4ac }}{2a}$

Discriminant: $D=\sqrt{ b^{2}-4ac }$ determines the classes of PDEs

$D>0$
$D=0$
$D<0$

Water Ripples

Which PDE can describe this phenomenon?

Hyperbolic
Parabolic
Elliptic

Dye in Agar

Which PDE can describe this phenomenon?

Hyperbolic
Parabolic
Elliptic

Can you give an example of an elliptic phenomenon?

Problems

Question 1

The lines along which a partial differential equation reduces to an ordinary differential equation are called

parabolic lines
hyperbolic lines
characteristic lines
None of these

Solution: Characteristic lines Consider $u_{t}+u_{x x}=0$ what are the characteristic lines for the above equation? $\frac{dt}{dx}=0$

Consider $u_{tt}-u_{x x}=0$ $a=-1,b=0,c=1$ $b^{2}-4ac=4>0$ $\frac{dt}{dx}=\pm{1}$

Question 2

Which of these apply to parabolic equations?

They have one real characteristic line
They have two real characteristic lines
They have two imaginary characteristic lines
They do not have characteristic lines

Solution: They have one real characteristic line

Question 3

Which of these characteristics apply to hyperbolic equation?

finite domain of dependence and infinite domain of influence
finite domain of dependence and finite domain of influence
no domain of dependence and no domain of influence
None of the above

Solution: finite domain of dependence and finite domain of influence

Question 4

The nature of the second order partial differential equation $\frac{ \partial \phi }{ \partial t }+\alpha \frac{ \partial \phi }{ \partial x }+\beta \frac{ \partial^{2} \phi }{ \partial x^{2} }=0$

circular
elliptic
parabolic
hyperbolic

Solution: parabolic

$a=\beta,b=0,c=0$ $D=b^{2}-4ac=0-\beta \times 0 \times 4=0$

Question 5

The value of $\alpha$ for which the equation $\alpha \frac{ \partial^{2}\phi }{ \partial x^{2} }+4 \frac{ \partial^{2}\phi }{ \partial x \partial y} + 2 \frac{ \partial^{2} \phi }{ \partial y^{2} }=0$ becomes parabolic is

Solution: 2

$4^{2}-4\times 2 \times \alpha=0 \implies \alpha=?$

Question 6

The steady compressible flow defined by $(1-M_{\infty}^{2})\frac{ \partial^{2}\phi }{ \partial x^{2} }+\frac{ \partial^{2}\phi }{ \partial y^{2} }=0$ becomes hyperbolic under the condition

$M_{\infty}=0$
$M_{\infty}=1$
$M_{\infty}>1$
$M_{\infty}<1$

Solution: $M_{\infty}>1$

$0-4(1-M_{inf}^{2})>0$ $\implies$ $M_{inf}>1$

Question 7

The Tricomi equation $y \frac{ \partial^{2} u }{ \partial x^{2} }+\frac{ \partial^{2}u }{ \partial y^{2} }=0$ shows elliptic nature under the condition

$y>0$
$y<0$
$y=0$
cannot be determined

Solution: $y>0$

$0-4y < 0$ $\implies$ $y>0$

Question 8

The following system of equations is classified as $\begin{align} \frac{ \partial u }{ \partial t } +8 \frac{ \partial v }{ \partial x } & =0 \\ \frac{ \partial v }{ \partial t }+2 \frac{ \partial u }{ \partial x } & = 0 \end{align}$

elliptic
parabolic
hyperbolic
None of the above

Solution: hyperbolic

2D NS : $u,v,p$ continuity x & y momentum $u_{x}+v_{y}=0$

$U = \begin{pmatrix}u \\ v\end{pmatrix}$ , $U_{x}=\begin{pmatrix}u_{x} \\ v_{x}\end{pmatrix}$ $U_{t}+AU_{x}=0$ $A=\begin{pmatrix}0 & 8 \\ 2 & 0\end{pmatrix}$

First eqn wrt t and then second eqn wrt x $u_{t}+8v_{x}=0$ $u_{tt}+8v_{xt}=0$

$v_{t}+2u_{x}=0$ $v_{tx}+2u_{x x}=0$

$v_{tx}=v_{xt}$ $u_{tt}-16u_{x x}=0$ $0-4 \times -16>0$ $\implies$ hyperbolic

Calculate eigen values for matrix A $\lambda^{2}-8\times{2}=0$

Question 9

The nature of the following Cauchy-Reimann equations is $\begin{align} \frac{ \partial u }{ \partial x } + \frac{ \partial v }{ \partial y } & =0 \\ \frac{ \partial v }{ \partial x } -\frac{ \partial u }{ \partial y } & =0 \end{align}$

elliptic
parabolic
hyperbolic
None of the above

Solution: elliptic

$U=\begin{pmatrix}u \\ v\end{pmatrix}$

$U_{x}+ A U_{y}=0$ where $U_{x}=\begin{pmatrix}u_{x} \\ v_{x}\end{pmatrix}$ and $U_{y}=\begin{pmatrix}u_{y} \\ v_{y}\end{pmatrix}$

$A=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$

characteristic equation for matrix $A$ : $\lambda^{2}+1=0$ eigen values for $A$ are $\pm i$

Question 10

The following system of equations is classified as ( $\beta$ has real value) $\begin{align} \beta^{2} \frac{ \partial u }{ \partial x } -\frac{ \partial v }{ \partial y } & =0 \\ \frac{ \partial v }{ \partial x } - \frac{ \partial u }{ \partial y } & =0 \end{align}$

elliptic
parabolic
hyperbolic
None of the above

Solution: hyperbolic

$U=\begin{pmatrix}u \\ v\end{pmatrix}$

$U_{x}+ A U_{y}=0$ where $U_{x}=\begin{pmatrix}u_{x} \\ v_{x}\end{pmatrix}$ and $U_{y}=\begin{pmatrix}u_{y} \\ v_{y}\end{pmatrix}$

$\begin{pmatrix} u_{x} \\ v_{x} \\ \end{pmatrix} + \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} u_{y} \\ v_{y} \end{pmatrix}=0$

$A=\begin{pmatrix}0 & -\frac{1}{\beta^{2}} \\ -1 & 0\end{pmatrix}$

$\lambda^{2}-\frac{1}{\beta^{2}}=0$

$\beta \in \mathbb{R}$ $\implies$ $\lambda$ $\in \mathbb{R}$ so hyperbolic

Thank You

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