The document discusses finite difference methods for approximating derivatives from discrete data points. It provides equations for first, second, and higher order forward, backward, and central finite difference approximations of derivatives. It also discusses applying these methods to mixed derivatives and using different orders of finite differences to calculate the shear stress on a flat plate from velocity data at discrete y-locations, comparing the results to the exact value.

1. Finite Differences
 If ui, j denotes the x component of velocity at point (i, j), then
the velocity ui + 1, j at point (i + 1, j) can be expressed in terms
of a Taylor series expanded about point (i, j) as follows:
 The first term on the right side is a finite difference
representation of the partial derivative. The remaining terms on
the right side constitute the truncation error. That is, if we wish
to approximate the derivative with first order forward
difference technique we get
L.K.SAHA 80

2. Finite Differences
 For first order accurate backward difference we get
 Second order accurate central difference
 For higher order derivatives
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3. Finite Differences
 fourth-order-accurate central finite-difference for 2u/ x2 is
 For mixed derivatives
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4. L.K.SAHA 83

5. L.K.SAHA 84

6. Finite Differences
Consider the viscous flow of air over a flat plate. At a given station in the flow
direction, the variation of the flow velocity, u, in the direction perpendicular to
the plate (the y direction) is given by the expression
Where L= characteristics length 3cm, the units of u meters per seconds. The
viscosity coefficient μ is 3.7373X 10-5 kg/(m-s). Using the above equation to
provide the values of u at discrete grid points equally spaced in the y direction,
with Δy=0.1cm. Using these discrete values, calculate the shear stress at the
wall w three different ways, namely:
i) first order forward difference
ii) second order forward difference
iii) third order forward difference
iv) Finally, compare these calculated finite-difference results with the exact
value of w.
L.K.SAHA 85
y, cm u,m/s
0 0
0.1 150.54
0.2 286.77
0.3 410.03