Aerodynamic Fundamentals

Aerodynamic Fundamentals

1. In the aerodynamic field, we like to know in which direction the flow is going, since in this way all the physical phenomena involved in it are investigated

   • Pathlines , streamlines , and streaklines of a flow
     • Pathlines
       • the line traced by a given particle as it flows from one point to another
         • its respective equation
           • dẊ/ dt (t) = up(Ẋp(t),t) Ẋp(to)=Ẋpo
             • P: indicates that we are following the motion of a fluid particle
             • Ẋp: the curve is parallel to the velocity vector
             • t :the speed evaluated in a certain time
     • streamlines
       • is a curve whose tangent at any point is in the direction of the velocity vector at that point
         • its respective equation
           • ds X V = 0
             • ds: a directed element of the streamline
             • V:is the speed parallel to the current line
             • Expanding it in Cartesian coordinates
               • ds = dx(i) + dy(j) + dz(k)
               • V= ui + vj + wk
                 • Making Cross Product
                   • 0=i(w dy - v dz) + j(u dz- w dx) + k(v dx- u dy)
                     • decomposing by each component
                       • w dy - v dz=0 u dz - w dx=0 v dx - udy=0
     • streaklines
       • a line created by particles in a flow that have previosly passed through a common point
         • its respective equation
           • dẊstr/ dt = up(Ẋstr,t) Ẋstr(t=τp)=Ẋpo
             • Ẋstr :speed of a particle at location and time
     • is the distinction between path lines and stream lines
   • Angular Velocity , Vorticity , and Strain
     • element orientation and its shape change as it moves along a streamline
     • Vorticity
       • they are the most powerful quantities in theoretical aerodynamics to quantify the amount of rotation and distortion.
         • Terms that compose it
           • • v△t
               • Distance in y direction that A moves during time increment △t
             • (v + au/ax dx) △t
               • Distance in y direction that C moves during time increment △t
             • (v + au/ax dx) △t - V△T
               • Net displacement in y direction of C relative to A
           • ξ=2ω
             • Cartesian coordinates
               • ξ=Ṽ X V
                 • ≠ 0
                   • Rotacional
                 • = 0
                   • Irrotacional
     • Strain
       • is the change in x, where the positive strain corresponds to a decreasing one in the x axis.
         • Terms that compose it
           • Strain= -△k=△θ2-△θ1
     • Angular Velocity
       • he average of the angular velocities of the lines are taken from one point to another point
         • Terms that compose it
           • • ωz=1/2(dθ1/dt + dθ2/dt)
               • n three-dimensional terms
                 • ω=ωi + ωj + ωk
                   • the angular velocity adds all its components in the velocity field
               • ω= angular velocity
             • dθ= number of angular changes
   • Circulation
     • The circulation is simply the negative of the line integral of velocity around a closed curve in the flow
       • its respective equation
         • • the integral is infinite
           • V and ds :is the velocity and the line segment
   • Stream Function
     • whose tangent at any point has the direction of the fluid velocity at that point
       • its respective equation
         • Ψ(x,y)=c
           • comes from the derivation
             • dl x V = 0
           • graphically
             • • By definition
                 • is the change obtained for each constant of the
               • △Ψ=C2-C1
           • For incompressible flows
             • ξz= - V^2 Ψ
           • For irrotational, incompressible flows
             • o=- V^2 Ψ
   • Velocity Potential
     • is a flow where the vorticity is zero at all points
       • ξ = V x v =o
         • Scalar Function
           • V x ( Vθ) =0
             • cartesian coordinates
               • u=аθ/аx v=аθ/аy w=аθ/аz
                 • Velocity potential is defined for irritation flow only
                 • The velocity potential is applied to three-dimensional flows
     • Relationship

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