AE4-135 Rotor/Wake Aerodynamics

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Biot-Savart law - velocity induced by a volume of vorticity.

The velocity $\vec{u_\omega}$ at a point $x_p$ associated with the existing vorticity $\vec{\omega}$ can be calculated using the Bio-Savart law, expressed by the integral below.

$\vec{u_\omega}=\frac{1}{4 \pi} \int\limits_V \frac{\vec{\omega}\times \vec{r}}{\left| r^3 \right|} \rm{d}V$

Press bottom arrow for Biot-Savart law applied to a vortex filament

Biot-Savart law - velocity induced by a vortex curve.

If the vorticity is concentrated in a curve from a point $a$ to a point $b$ with constant circulation $\Gamma$ , the velocity $\vec{u_\omega}$ at a point $x_p$ can be calculated by the integral below.

$\vec{u_\omega}=\frac{\Gamma}{4 \pi} \int_a^b \frac{\vec{\rm{d}l}\times \vec{r}}{\left| r^3 \right|}$

The curve can be discretized in straight segments. If the vortex segment is straight, the integral below becomes a simple geometric relation.

Press right arrow for Biot-Savart law applied to a vortex filament

Velocity induced by a straight vortex filament

Change values below to change the direction and magnitude of the vortex filament. Change the location of the point to evaluate the induced velocity $X_p$ .

Press bottom arrow for algorithm to calculate the velocity induced by a straight vortex filament.

Algorithm for velocity induced by vortex filament

Consider a vortex filament from point $X_1$ to point $X_2$ , with strength $\Gamma$ . The velocity at a point $X_p$ can be calculated by the algorithm below.

$R_1= \sqrt{ (X_P-X_1)^2 + (Y_P-Y_1)^2 + (Z_P-Z_1)^2}$

$R_2= \sqrt{ (X_P-X_2)^2 + (Y_P-Y_2)^2 + (Z_P-Z_2)^2}$

${R_{1-2}}_X=(Y_P-Y_1)(Z_P-Z_2)-(Z_P-Z_1)(Y_P-Y_2)$

${R_{1-2}}_Y=-(X_P-X_1)*(Z_P-Z_2)+(Z_P-Z_1)*(X_P-X_2)$

${R_{1-2}}_Z=(X_P-X_1)*(Y_P-Y_2)-(Y_P-Y_1)*(X_P-X_2)$

${R_{1-2}}_{sqr}={{R_{1-2}}_X}^2 + {{R_{1-2}}_Y}^2 + {{R_{1-2}}_Z}^2$

$R_{0-1} = (X_2-X_1)(X_P-X_1)+(Y_2-Y_1)(Y_P-Y_1)+(Z_2-Z_1)(Z_P-Z_1)$

$R_{0-2} = (X_2-X_1)(X_P-X_2)+(Y_2-Y_1)(Y_P-Y_2)+(Z_2-Z_1)(Z_P-Z_2)$

$K=\frac{\Gamma}{4 \pi {R_{1-2}}_{sqr}} \left( \frac{R_{0-1}}{R_1}- \frac{R_{0-2}}{R_2} \right)$

$U=K*{R_{1-2}}_X$ ; $V=K*{R_{1-2}}_Y$ ; $W=K*{R_{1-2}}_Z$

Source: Katz, Joseph, and Allen Plotkin. Low-speed aerodynamics. Vol. 13. Cambridge university press, 2001.

Press bottom arrow for example of code implementation of this algorithm.

Code for calculation of velocity induced by a straight vortex filament

                      

                        // 3D velocity induced by a vortex filament
                        function velocity_3D_from_vortex_filament(GAMMA,XV1, XV2, XVP1,CORE){
                          // function to calculate the velocity induced by a straight 3D vortex filament
                          // with circulation GAMMA at a point VP1. The geometry of the vortex filament
                          // is defined by its edges: the filament starts at XV1 and ends at XV2.
                          // the input CORE defines a vortex core radius, inside which the velocity
                          // is defined as a solid body rotation.
                          // The function is adapted from the algorithm presented in:
                          //                Katz, Joseph, and Allen Plotkin. Low-speed aerodynamics.
                          //                Vol. 13. Cambridge university press, 2001.

                          // read coordinates that define the vortex filament
                          var X1 =XV1[0]; var Y1 =XV1[1]; var Z1 =XV1[2]; // start point of vortex filament
                          var X2 =XV2[0]; var Y2 =XV2[1]; var Z2 =XV2[2]; // end point of vortex filament
                          // read coordinates of target point where the velocity is calculated
                          var XP =XVP1[0]; var YP =XVP1[1]; var ZP =XVP1[2];
                          // calculate geometric relations for integral of the velocity induced by filament
                          var R1=Math.sqrt(Math.pow((XP-X1), 2) + Math.pow((YP-Y1), 2) + Math.pow((ZP-Z1), 2) );
                          var R2=Math.sqrt( Math.pow((XP-X2), 2) + Math.pow((YP-Y2), 2) + Math.pow((ZP-Z2), 2) );
                          var R1XR2_X=(YP-Y1)*(ZP-Z2)-(ZP-Z1)*(YP-Y2);
                          var R1XR2_Y=-(XP-X1)*(ZP-Z2)+(ZP-Z1)*(XP-X2);
                          var R1XR2_Z=(XP-X1)*(YP-Y2)-(YP-Y1)*(XP-X2);
                          var R1XR_SQR=Math.pow(R1XR2_X, 2)+ Math.pow(R1XR2_Y, 2)+ Math.pow(R1XR2_Z, 2);
                          var R0R1 = (X2-X1)*(XP-X1)+(Y2-Y1)*(YP-Y1)+(Z2-Z1)*(ZP-Z1);
                          var R0R2 = (X2-X1)*(XP-X2)+(Y2-Y1)*(YP-Y2)+(Z2-Z1)*(ZP-Z2);
                          // check if target point is in the vortex filament core,
                          // and modify to solid body rotation
                          if (R1XR_SQR < Math.pow(CORE,2)) {
                            R1XR_SQR=Math.pow(CORE,2);
                            // GAMMA = 0;
                          };
                          if (R1 < CORE) {
                            R1=CORE;
                            // GAMMA = 0;
                          };
                          if (R2 < CORE) {
                            R2=CORE;
                            // GAMMA = 0;
                          };
                          // determine scalar
                          var K=GAMMA/4/Math.PI/R1XR_SQR*(R0R1/R1 -R0R2/R2 );
                          // determine the three velocity components
                          var U=K*R1XR2_X;
                          var V=K*R1XR2_Y;
                          var W=K*R1XR2_Z;
                          // output results, vector with the three velocity components
                          var results = [U, V, W];
                          return(results) };

Discretization of each horseshoe

Each horseshoe is discretized in straight vortex filaments. The blade elements are discretized by three filaments: one bound at the quarter-chord line, two trailing in chord direction. According to Helmholtz theorem, the circulation $\Gamma$ is constant for all filaments that compose the horseshoe. The direction of the filament defines the direction of the circulation.

Press bottom arrow for further information on "Discretization and grid generation".

Discretization and geometry of the fixed wake

When defining the geometry of the fixed wake, you need to define the convection and expansion of the wake. We will only consider an axial convection velocity $U_{wake} = U_\infty \left( 1-a_w \right)$ . Assume in a first approach that $a_w$ is equal to the average induction at the rotor.

Consider that the wake is aligned in axial direction $x$ . The location of the nodes of a filament released from a radial position $r$ , at a given past time of $t$ , is defined by:

$x_w = t \cdot U_{wake}$

$y_w = r \cdot \sin \Omega t$

$z_w = r \cdot \cos \Omega t$

Scroll the button below to vary the axial convection speed. $a_w$ = 0.20

Setting matrices for induced velocity by vortex system

Potential flow allows for a linear solution of the velocity field. Therefore, we can define a linear system to calculate the velocity induced by all horseshoe vortex rings at all control points.

Consider that we have a total of $N$ blade sections with $N$ controlpoints, releasing $N$ horseshoe vortex rings. We define $i_{cp}$ as the index of a control point at a blade element, and $j_{ring}$ as the index of the vortex ring.

We now define the velocity "induced" by a horseshoevortex ring $j_{ring}$ at point $i_{cp}$ as $\vec{u}_{i_{cp}-j_{ring}}$

The three velocity components of $\vec{u}_{i_{cp}-j_{ring}}$ in $x$ , $y$ and $z$ -direction are $u_{i_{cp}-j_{ring}}$ , $v_{i_{cp}-j_{ring}}$ and $w_{i_{cp}-j_{ring}}$

Because the velocity induced by a vortex is linear with the strength of the vortex, $\vec{u}_{i_{cp}-j_{ring}} = \vec{u}_{i_{cp}-j_{ring}|_{\Gamma=1}}*\Gamma_{j_{ring}}$ , where $\vec{u}_{i_{cp}-j_{ring}|_{\Gamma=1}}$ is the velocity induced by a horseshoe vortex of unit strength.

The total velocity by all horseshoe vortices at one controlpoint $\vec{u}_{i_{cp}}$ is given by $\vec{u}_{i_{cp}}=\sum_{j_{ring}=1}^N \vec{u}_{i_{cp}-j_{ring}} =\sum_{j_{ring}=1}^N \vec{u}_{i_{cp}-j_{ring}|_{\Gamma=1}}*\Gamma_{j_{ring}}$

Press bottom arrow to see remaining formulation of velocity components.

Vector formulation of velocity components induced at one controlpoint by N horseshoe vortex rings

${u}_{i_{cp}} = \left[ {\begin{array}{*{20}{c}} {u}_{i_{cp}-1|_{\Gamma=1}} & \cdots & {u}_{i_{cp}-N|_{\Gamma=1}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} \Gamma_1 \\ \vdots \\ \Gamma_N \end{array}} \right]$ ${v}_{i_{cp}} = \left[ {\begin{array}{*{20}{c}} {v}_{i_{cp}-1|_{\Gamma=1}} & \cdots & {v}_{i_{cp}-N|_{\Gamma=1}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} \Gamma_1 \\ \vdots \\ \Gamma_N \end{array}} \right]$ ${w}_{i_{cp}} = \left[ {\begin{array}{*{20}{c}} {w}_{i_{cp}-1|_{\Gamma=1}} & \cdots & {w}_{i_{cp}-N|_{\Gamma=1}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} \Gamma_1 \\ \vdots \\ \Gamma_N \end{array}} \right]$

Press bottom arrow to see matrix formulation.

Matrix formulation of velocity components induced at $N$ controlpoint by $N$ horseshoe vortex rings

$\left[ {\begin{array}{*{20}{c}} u_1\\ \vdots \\ u_N \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {u}_{1-1|_{\Gamma=1}} & \ldots & {u}_{1-N|_{\Gamma=1}}\\ \vdots & \ddots & \vdots \\ {u}_{N-1|_{\Gamma=1}} & \ldots & {u}_{N-N|_{\Gamma=1}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} \Gamma_1 \\ \vdots \\ \Gamma_N \end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}} v_1\\ \vdots \\ v_N \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {v}_{1-1|_{\Gamma=1}} & \ldots & {v}_{1-N|_{\Gamma=1}}\\ \vdots & \ddots & \vdots \\ {v}_{N-1|_{\Gamma=1}} & \ldots & {v}_{N-N|_{\Gamma=1}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} \Gamma_1 \\ \vdots \\ \Gamma_N \end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}} w_1\\ \vdots \\ w_N \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {w}_{1-1|_{\Gamma=1}} & \ldots & {w}_{1-N|_{\Gamma=1}}\\ \vdots & \ddots & \vdots \\ {w}_{N-1|_{\Gamma=1}} & \ldots & {w}_{N-N|_{\Gamma=1}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} \Gamma_1 \\ \vdots \\ \Gamma_N \end{array}} \right]$

Calculate velocities, circulations and forces at the blade element

Velocity perceived at blade element: axial, tangential and total.

$V_{axial} = U_\infty + u$
$V_{tan} = \Omega r + \left[ {\begin{array}{*{20}{c}} U_\infty + u & v & w \end{array}} \right] \cdot \vec{n}_{azim}$
$V_{p} = \sqrt{V_{axial}^2 + V_{tan}^2 }$

Circulation and forces at blade element (two dimensional): lift, drag, axial and azimuthal.

$\Gamma = \frac{1}{2} c V_{p} C_{l{\left(\alpha\right)}}$
$Lift = \frac{1}{2} c \rho V_{p}^2 C_{l{\left(\alpha\right)}}$
$Drag = \frac{1}{2} c \rho V_{p}^2 C_{d{\left(\alpha\right)}}$
$F_{azim} = Lift \sin{\Phi}- Drag \cos{\Phi}$
$F_{axial} = Lift \cos{\Phi} + Drag \sin{\Phi}$

Note that all forces are in 2D, and have dimensions $N/m$ .

$C_l$ and $C_d$ can be retrieved from airfoil polar data, as a function of angle of attack $\alpha$ .

Press bottom arrow to see an example of an airfoil polar.

Example of Lifiting Line solution of bound circulation and comparison with BEM solution.

Scroll the button below to change tip speed ratio. Tip speed ratio = 6.00

Rotor performance:

	BEM	Lifting Line
$C_T$	0.49	0.48
$C_P$	0.39	0.39

Note: The circulation $\Gamma$ is non-dimensioned by $\frac{ U_\infty^2 \pi }{N_{Blades} \Omega}$

Press bottom arrow to see other examples.

Example of Lifting Line solution of $F_{axial}$ and $F_{azim}$ , and comparison with BEM solution.

Scroll the button below to change tip speed ratio. Tip speed ratio = 6.00

Rotor performance:

	BEM	Lifting Line
$C_T$	0.49	0.48
$C_P$	0.39	0.39

Note: $F_{axial}$ and $F_{azim}$ are non-dimensioned by $\frac{1}{2} \rho U_\infty^2 R$

Press bottom arrow to see other examples.

Example of Lifting Line solution of $a$ and $a'$ , and comparison with BEM solution.

Scroll the button below to change tip speed ratio. Tip speed ratio = 6.00

Rotor performance:

	BEM	Lifting Line
$C_T$	0.49	0.48
$C_P$	0.39	0.39

Effect of the number of blade sections.

Scroll the button below to change number of spanwise elements. N = 16

Rotor performance:

	BEM	Lifting Line
$C_T$	0.49	0.48
$C_P$	0.39	0.39

Note: The circulation $\Gamma$ is non-dimensioned by $\frac{ U_\infty^2 }{N_{Blades} \pi \Omega}$ . Solution at tip speed ratio = 6.

Press bottom arrow to see other examples.

Effect of length of the wake.

Scroll the button below to vary the length of the wake $Lw$ in diameters D downstream. Lw/D = 1.29

Rotor performance:

	BEM	Lifting Line
$C_T$	0.49	0.49
$C_P$	0.39	0.40

Lifting line model

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Learning objectives

Nomenclature

Concept of the lifting line model

Flowchart of a lifting line model

Steps for setup and solution of the lifting line model.

Choice of the vortex filament as perturbation element.

Biot-Savart law - velocity induced by a volume of vorticity.

Biot-Savart law - velocity induced by a vortex curve.

Velocity induced by a straight vortex filament

Algorithm for velocity induced by vortex filament

Code for calculation of velocity induced by a straight vortex filament

Setup geometry of the horseshoe vortices

Discretization of each horseshoe

Spanwise discretization

Discretization and geometry of the fixed wake

Setting matrices for induced velocity by vortex system

Vector formulation of velocity components induced at one controlpoint by N horseshoe vortex rings

Matrix formulation of velocity components induced at $N$ controlpoint by $N$ horseshoe vortex rings

Calculate velocities, circulations and forces at the blade element

Example of airfoil polar, with lift and drag coefficient as a function of angle of attack $C_{l{\left(\alpha\right)}}$ and $C_{d{\left(\alpha\right)}}$

Example of lifting line code implementation - solution of the matrix system

Example of Lifiting Line solution of bound circulation and comparison with BEM solution.

Example of Lifting Line solution of $F_{axial}$ and $F_{azim}$ , and comparison with BEM solution.

Example of Lifting Line solution of $a$ and $a'$ , and comparison with BEM solution.

Effect of discretization of the vorticity system

Effect of the number of blade sections.

Effect of length of the wake.

Suggestion: the example of the rectangular wing in straight flight

Review of tutorial

Learning objectives

Suggestions