Aerodynamics

1) The temperature and pressure at the stagnation point of a high speed missile are \({934^ \circ }\) R and 7.8 atm, respectively.Calculate the density at this point. Solution: \[\begin{array}{l}T = {934^ \circ }R\\p = 7.8\,atm\\Density = \rho  = ?\\p = \rho RT\\\rho  = \frac{P}{{RT}} = \frac{{\left( {7.8 \times 2116} \right)}}{{1716 \times 934}} = 0.0103\,slug/f{t^3}\end{array}\]2)Calculate \({c_p},{c_v},e\,{\rm{and}}\,h\) for  a) The stagnation point conditions given in problem (1). b) Air at standard sea level conditions. Solution: \({c_p},{c_v},e\,{\rm{and}}\,h\) for stagnation point conditions will be  a)  \[{c_p} = \frac{{\gamma R}}{{\gamma  - 1}} = \frac{{1.4 \times 1716}}{{0.4}} = 6006\frac{{ft}}{{slug}}\frac{{lb}}{{{}^ \circ R}}\]\[{c_v} = \frac{R}{{\gamma  - 1}} = \frac{{1716}}{{0.4}} = 4290\frac{{ft}}{{slug}}\frac{{lb}}{{{}^ \circ R}}\]\[e = {c_v}T = 4290(934) = 4.007 \times {10^6}\frac{{ft\,lb}}{{slug}}\]\[h ...

Euler's equation is given as \[dp =  - \rho vdv\]This equation applies to an incompressible and in-viscid flow where \(\rho  = \,{\rm{constant}}\). In a streamline, in between two points 1 and 2, the above Euler equation can be integrated as \[\int\limits_{{p_1}}^{{p_2}} {dp =  - \rho \int\limits_{{v_1}}^{{v_2}} {vdv} } \]\[{p_2} - {p_1} =  - \rho \left( {\frac{{v_2^2}}{2} - \frac{{v_1^2}}{2}} \right)\]\[ \Rightarrow {p_1} + \frac{1}{2}\rho v_1^2 = {p_2} + \frac{1}{2}\rho v_2^2\]This can be written as\[p + \frac{1}{2}\rho {v^2} = {\rm{constant}}\] for a streamline.For an rotational flow the value of constant is changing from streamline to another.For irrotational flow,the constant is same for all streamlines and \[p + \frac{1}{2}\rho {v^2} = {\rm{constant}}\]throughout the flow. Physical significance of Bernoulli's equation is that when the velocity increases, the pressure decreases  and when the velocity decreases, the pressure increases. ...

When there is a superposition of a uniform flow with a doublet (Which is a source-sink pair of flows), a non-lifting flow over a circular cylinder can be analysed. Stream function \('\psi '\) for a uniform flow is \[\psi  = \left( {{V_\infty }r\sin \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\] velocity field is obtained by differentiating above equation\[{V_r} = \left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\cos \theta \]\[\,{\rm{and}}\,{{\rm{V}}_\theta } =  - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta \] Stagnation points can be obtained by equating \({V_r}\) and \({{\rm{V}}_\theta }\) to zero.Considering an incompressible and inviscid flow pressure coefficient over a circular cylinder is \[{C_p} = 1 - 4{\sin ^2}\theta \]Example: Calculate locations on the surface of a cylinder where surface pressure equals to free-stream pressure considering a non-lifting flow. Solution: Pressure coefficient ...

Lifting flow over a circular cylinder can be synthesised as superposition of a vortex flow of strength \('\tau '\) with non lifting flow over a cylinder. The stream function for the flow can be given as\[\psi  = \left( {{V_\infty }r\sin \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right) + \frac{\tau }{{2\pi }}\ln \frac{r}{R}\]and velocity components in the radial and tangential direction can be obtained as \[{V_r} = \left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\cos \theta \] \[{V_\theta } =  - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta  - \frac{\tau }{{2\pi r}}\]The velocity on the surface of the cylinder can be given as  \[V =  - 2{V_\infty }\sin \theta  - \frac{\tau }{{2\pi R}}\] and coefficient of lift as \[{c_l} = \frac{\tau }{{R{V_\infty }}}\]Lift per unit span is given as \[{L^\prime} = {\rho _\infty }{V_\infty }\tau \] Example: Calculate the peak coefficient of pressure for a...

This theorem states that the lift per unit span is directly proportional to circulation.\[{L^\prime} = {\rho _\infty }{V_\infty }\tau \]Example: Lift per unit span of a spinning circular cylinder in a free stream with velocity  of 50 m/s is 8 N/m at a standard sea level conditions.Find the circulation \('\tau '\) around the cylinder. Solution: Lift per unit span is given as \({L^\prime} = {\rho _\infty }{V_\infty }\tau \)\[\tau  = \frac{{{L^\prime}}}{{{\rho _\infty }{V_\infty }}} = \frac{8}{{\left( {1.23} \right)\left( {50} \right)}} = 0.13\,{m^2}/s\]

For an incompressible flow \(\rho  = {\rm{constant}}\). \(\nabla  \cdot V\) is physically the time rate of change of volume of a moving fluid element per unit volume. For an incompressible flow, volume of a fluid element is constant, therefore \[\nabla  \cdot V = 0\]This can be also shown from continuity equation. Continuity equation is given as\[\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot \rho V = 0\]Since, for an incompressible flow, \(\rho  = {\rm{constant}}\) we have \[\frac{{\partial \rho }}{{\partial t}} = 0\]therefore the continuity equation becomes \[0 + \nabla  \cdot \rho V = 0\]\[\nabla  \cdot V = \frac{0}{\rho } = 0\]For an irrotational flow, velocity potential \('\phi '\) is defined as \[V = \nabla \phi \]Therefore, a flow which is both incompressible and irrotational the equation can be written as \[\nabla .\left( {\nabla \phi } \right) = 0\]\[{\nabla ^2}\phi  = 0\]This equation...

Pressure coefficient Cp is defined as \[{C_p} = \frac{{p - {p_\infty }}}{{{q_\infty }}}\]\({q_\infty } = \frac{1}{2}{\rho _\infty }v_\infty ^2\), let for any point in the flow where pressure and velocity are 'p' and 'v', respectively and free-stream pressure and velocity be \({{p_\infty }}\) and \({v_\infty }\). From Bernoulli's equation\[{p_\infty } + \frac{1}{2}\rho v_\infty ^2 = p + \frac{1}{2}\rho {v^2}\]\[ \Rightarrow \left( {p - {p_\infty }} \right) = \frac{1}{2}\rho \left( {v_\infty ^2 - {v^2}} \right)\]\[{C_p} = \frac{{p - {p_\infty }}}{{{q_\infty }}} = \frac{{\frac{1}{2}\rho \left( {v_\infty ^2 - {v^2}} \right)}}{{\frac{1}{2}\rho v_\infty ^2}}\]\[ \Rightarrow {C_p} = 1 - {\left( {\frac{v}{{{v_\infty }}}} \right)^2}\]This equation is valid for incompressible flow only. Example: Find pressure coefficient at a point on an  airfoil where velocity is 220 ft/s, which is in a free stream flow of 100 ft/s. Solution: pressure coefficient is given as \[...

Continuity equation is used to describe the transport of some quantities. It is based on the law of conservation of mass which states that mass can neither be created nor destroyed. For a flow process through a control volume where the stored mass does not change, fluid enters and leaves the controlled volume through its surface called controlled  surface. Inflow of fluid equals to outflow which is net mass flow out of control volume through surface S = time rate of decrease of mass inside control volume \(v\).\[\frac{\partial }{{\partial t}}\mathop{{\int\!\!\!\!\!\int\!\!\!\!\!\int}\mkern-31.2mu \bigodot}\limits_v  {\rho dv + \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\limits_S  {\rho v.ds = 0} } \] Continuity equation in the form of partial differential equation is \[\frac{{\partial \rho }}{{\partial t}} + \nabla .\left( {\rho u} \right) = 0\] For a steady flow,\[\frac{\partial }{{\partial t}} = 0\]\[\nabla  \cdot \left( {\rho v} \right) = 0\]...

Example: Calculate the location of center of pressure of a airfoil section at an angle of attack of  \({{\rm{5}}^{\rm{o}}}\) , whose coefficient of lift,  \({{\rm{c}}_{\rm{l}}}\)  is 0.80 and  \({{\rm{c}}_{{\rm{m,c/4}}}}\)  is -0.08.Consider incompressible flow over the airfoil. Solution:- Location of center of pressure is given as \[{x_{cp}} = \left( {\frac{c}{4}} \right) - \frac{{M_{c/4}^\prime }}{{{L^{^\prime }}}}\] Where\({x_{cp}}\)  is the distance of center of pressure from the leading edge of airfoil, and 'c' is the chord length .\(M_{c/4}^\prime \) is moment per unit span about quarter-chord point and \({L^\prime }\) is the lift per unit span;\[\begin{array}{l}{x_{cp}} = \frac{c}{4} - \frac{{M_{c/4}^\prime }}{{{L^\prime }}}\\\frac{{{x_{cp}}}}{c} = \frac{1}{4} - \frac{{\left( {M_{c/4}^\prime /{q_\infty }{c^2}} \right)}}{{\left( {{L^\prime }/{q_\infty }c} \right)}}\end{array}\]\[\begin{array}{l} = \frac{1}{4} - \left( {\frac{{{c_{...

Aerodynamic coefficients plays an important role in performance analysis as well as  design of aeroplanes.Coefficient of lift is defined as Lift divided by dynamic pressure, \(L/{q_\infty }S\) where as drag coefficient is defined as Drag divided by dynamic pressure, \(D/{q_\infty }S\) . Coefficient of lift,maximum or \({C_{L,\max }}\) is the determining factor for stalling velocity of aircraft.The higher is the \({C_{L,\max }}\) the lower is the stalling velocity.\[{V_{stall}} = \sqrt {\frac{{2W}}{{{\rho _\infty }S{C_{L,\max }}}}} \] However,\({C_{L,\max }}\) can be increased by the use of mechanical devices like high-lift devices.High lift devices include flaps,slats and slots on the wing. Air plane flying at given altitude with maximum thrust \({T_{\max }}\), the maximum value of \({V_\infty }\) ,corresponds to flight at \({C_{D,\min }}\).\[{V_{\max }} = \sqrt {\frac{{2{T_{\max }}}}{{{\rho _\infty }S{C_{D,\min }}}}} \] The actual value of \({C_L}\) and...

Pathlines: It is the path traced by a fluid particle over a given period of time.For unsteady flow,path lines for different fluid particles passing through the same point are not same.                                 Streamlines: Streamlines represents the curve where tangent at any point on the curve is in the direction of velocity vector, at that point.For an unsteady flow the streamline patterns changes at different times.                                          Streaklines: It is the locus of the fluid elements that have passed through a particular point over a period of time.      Pathlines, streamlines and streaklines are all same curves for a steady flow. For unsteady flow they are different.

Streamlines: Streamlines represents field lines in a fluid flow. It is a path traced out by massless fluid particles moving with the flow, which is tangential to the instantaneous velocity direction. Different streamlines at same instant in a flow do not intersect or flow across it, because a fluid particle cannot have two different velocities at the same point. For a unsteady flow, streamline pattern is different at different times because the velocity vectors are fluctuating with time both in magnitude and direction. In a fluid flow, a bundle of streamlines is called a streamtube. Walls of an ordinary garden hose form a streamtube for the water flowing through the hose. Differential equation for a streamline in two dimensions is \[vdx - udy = 0\] Question: x and y components of a velocity field are given as \(u = \frac{{4x}}{{\left( {{x^2} + {y^2}} \right)}}\) and \(v = \frac{{4y}}{{\left( {{x^2} + {y^2}} \right)}}\), what is the equation of streamlines.   ...

A flow is called rotational if \(\nabla  \times V \ne 0\)  at every point in a flow. The fluid element has a finite angular velocity. A flow field is called irrotational if \(\nabla  \times V = 0\) at every point in a flow. The fluid elements have no angular velocity and the motion is translation. Example: A velocity field has a radial component of velocity \({V_r} = 0\) and tangential components of velocity \({V_\theta } = 4r\), respectively. Is this flow rotational or irrotational?   Solution: Here  \({V_r} = 0\) and  \({V_\theta } = 4r\) \[\begin{array}{*{20}{l}}{\nabla  \times \mathop V\limits^ \to   = {e_z}\left[ {\frac{{\partial {V_\theta }}}{{\partial r}} + \frac{{{V_\theta }}}{r} - \frac{1}{r}\frac{{\partial \left( {{V_r}} \right)}}{{\partial \theta }}} \right]}\\{\nabla  \times \mathop V\limits^ \to   = {e_z}\left[ {\frac{{\partial \left( {4r} \right)}}{{\partial r}} + \frac{{4r}}{r} - \frac{1}{r}\frac{...

Circulation is the line integral around a closed curve of the velocity field.It is normally denoted by  \(\tau \).\[\tau  =  - \oint_c {V.ds} \] Circulation should not be confused by its general dictionary meaning, in aerodynamics it is the mathematical line integral around a closed curve of a velocity field - a technical term. Circulation is related to vorticity through stoke's theorem \[\tau  =  - \oint_c {V.ds}  =  - \iint\limits_s {\left( {\nabla  \times V} \right)}.ds\] Example: Consider a velocity field where x and y components of velocity are  \(u = 5y/\left( {{x^2} + {y^2}} \right)\) and  \(v =  - 5x/\left( {{x^2} + {y^2}} \right)\). Calculate circulation around a circular path of radius  \(3m\) . Let the units of  \(u\)  and  \(v\)  be in  \(m/s\) . Solution: Let  \(x = r\cos \theta ,\,y = r\sin \theta \) , therefore  \({x^2} + {y^2} = {r^2}\). In polar- coordi...

Vortex is a region in a fluid in which flow revolves around an axis line. Tornadoes form during thunderstorms when warm, humid air collides with colder air to form a vortex. A whirlpool is a body of swirling water produced by the meeting of opposite currents. Whirlpool having a downdraft is termed as vortex .  Vorticity defines the dynamics of vortices, by a vector that describes the local rotary motion at a point in the fluid. In a velocity field, the curl of velocity is equal to vorticity\[\xi  = \nabla  \times V\] i) If \(\nabla  \times V = 0\)  at every point in a flow, it is a irrotational flow. ii) If  \(\nabla  \times V \ne 0\) at every point in a flow, it is a rotational flow.   Example: Consider a velocity field where x and y components of velocity are \(u = 4y/\left( {{x^2} + {y^2}} \right)\) and \(v =  - 4x/\left( {{x^2} + {y^2}} \right)\).Calculate the vorticity. Solution: Vorticity is given as...

In a vortex flow streamlines are concentric circles about a given point.The velocity along any given streamline is constant and varies from one streamline to another inversely with distance from the common centre.The velocity components in radial and tangential directions are \[\begin{array}{l}{V_r} = 0\,{\rm{and}}\,\,{V_\theta } = \frac{{{\rm{constant}}}}{r}\\{V_\theta } = \frac{{ - \tau }}{{2\pi r}}\end{array}\] \(\tau \)  = strength of vortex flow. Example: For a vortex flow, \(u = \frac{{4x}}{{{x^2} + {y^2}}}\,and\,v = \frac{{ - 4y}}{{{x^2} + {y^2}}}\),calculate(a) The time rate of change of the volume of a fluid element per unit volume. (b) The vorticity. Solution:  On changing the equation to polar co-ordinates  \[\begin{array}{l}x = r\cos \theta \\y = r\sin \theta \\{V_r} = u\cos \theta  + v\sin \theta \\{V_\theta } =  - u\sin \theta  + v\cos \theta \end{array}\] \[\begin{array}{l}u = \frac{{4y}}{{({x^2} + {y^2})}} = \frac{{4r\sin \theta }}{{{...

Source flow:It is type of flow,where all the streamlines are straight lines emanating from a central point.The velocity along each of the streamlines vary inversely with distance from central point.Velocity components in the radial and tangential directions are \({V_r}\,{\rm{and}}\,{V_\theta }\).\[\begin{array}{l}{V_r} = \frac{\Lambda }{{2\pi r}}\\\Lambda  = {\rm{source}}\,{\rm{strength}}\\{V_\theta } = 0\end{array}\]Example:For a source flow ,calculate (a).The time rate of change of the volume of a fluid element per unit volume. (b) The vorticity. Solution : (a)\[\nabla .\mathop V\limits^ \to   = \frac{1}{{\partial V}}\frac{{D\left( {\partial V} \right)}}{{Dt}}\] In polar-coordinates we have \[\nabla .\mathop V\limits^ \to   = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{V_r}} \right) + \frac{1}{r}\frac{{\partial {V_\theta }}}{{\partial \theta }}\]Let \[\begin{array}{l}x = r\cos \theta \\y = r\sin \theta \\{V_r} = u\cos \theta  + v\sin \t...

Stream function \(\left( {{\rm{\psi  = constant}}} \right)\) gives the equation of a streamline and the flow velocity is obtained by differentiating \({\rm{\psi }}\).For a compressible flow\[\rho u = \frac{{\partial \psi }}{{\partial y}}\]\[\rho v =  - \frac{{\partial \psi }}{{\partial x}}\]For incompressible flow,\[u = \frac{{\partial \psi }}{{\partial y}}\]\[v =  - \frac{{\partial \psi }}{{\partial x}}\]Velocity potential: Velocity potential is defined by \(\phi ,\phi  = \phi \left( {x,y,z} \right)\).For an irrotational flow,velocity is given by gradient of \(\phi \).\[u = \frac{{\partial \phi }}{{\partial x}},v = \frac{{\partial \phi }}{{\partial y}},w = \frac{{\partial \phi }}{{\partial z}}\]Stream function is defined for both rotational and irrotational flows,velocity potential is defined for irrotational flows only.However,stream function is defined for two-dimensional flows only,the velocity potential applies to three-dimensional flows al...

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