The Gurney flap is a small plate attached to or near the trailing edge of an airfoil, perpendicular to the flow on the pressure side. Being counterintuitive to the idea of a streamlined body, the Gurney flap was invented more or less accidentally by race car driver Dan Gurney.
There are two basic (and somewhat overlapping) explanations for the lift-enhancing effect:
- The flap effectively increases camber [1], decelerating flow on the pressure side and accelerating flow on the suction side.
- The flap decelerates the flow on the pressure side right until the edge and leads to a pressure difference between suction and pressure side which in turn leads to an increased circulation [2].
However, the flow is rather complex and includes unsteady vortex-shedding[3] at the trailing edge.
To model the effects of a Gurney flap with viiflow, a single virtual displacement element is used depending on the position of the Gurney flap. The effect of Gurney flap position and height is analyzed and compared to the experimental results. As far as the author knows, this is the first numerical analysis of Gurney flaps using a panel method. The possibility of modeling the effect of a Gurney flap using a change in the Kutta condition of a panel method has been proposed in [2], but no function that correlates the Kutta condition (or a step in pressure) to the Gurney flap height is given.
The effect of Gurney flaps on the lift of an airfoil are surprisingly well predicted with viiflow. This does not hold for the drag, though. The drag increase for Gurney flaps that are not at the trailing edge can be computed with similar accuracy compared to a fully-turbulent RANS solution, but the coefficients were significantly overpredicted. For Gurney flaps right at the trailing edge, the drag cannot be computed using the wake boundary layer at this point, and the additional drag due to the Gurney flap is estimated using the local pressure at the flap.
Experiments¶
The most comprehensive experimental analysis to date on an airfoil can be found in [7] where the S903 airfoil has been analyzed. The coordinates of the airfoil can be found in [8], the paneling has been increased near the trailing edge. Gurney flap position, height and influence of transition location have been evaluated. In [4] some of these experimental results are compared with computational results with a fully-turbulent RANS solver. A note on these experimental results: the legend in the drag polar plot in [7], Fig. 6, is wrong. Here, the plots have been changed assuming decreasing drag at higher lift coefficients with decreasing chord position.
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
import viiflow as vf
import viiflowtools.vf_tools as vft
import viiflowtools.vf_plots as vfp
import matplotlib.pyplot as plt
import matplotlib
import numpy as np
import os
os.environ["OMP_NUM_THREADS"] = "1"
colors = ['k','k','k','k','k','k','k','k','k']
colors = ['tab:blue','tab:orange','tab:green','tab:red','tab:purple','tab:brown','tab:pink','tab:olive','tab:cyan']
marks = ['o','^','v','<','>','d','^','^','^']
Results¶
In all cases, viiflow accurately predicts the increase in lift and change in maximum lift. The drag coefficient is less accurately predicted.
Where the Gurney flap is right at the trailing edge, some parts of the increase in drag is invisible to viiflow without further modifications. In viiflow, the drag is calculated using the Squire-Young formula using the wake boundary layer development. The wake boundary layer development depends on its initialization at the trailing edge, which is done by adding the momentum and displacement thickness of suction and pressure side and a weighted average of the shear-stress. At this point, the modeling of the flow at the trailing edge is insufficient to describe the development of the wake behind a Gurney flap. However, the increase in drag seems to be largely dependent on the pressure difference at the Gurney flap [6]. Adding the pressure afterwards to the drag coefficient markedly improves the drag prediction. In fact, the resulting polar matches the experiments better than the results of a Gurney flap at .9c or .95c.
To compare the results an Ncrit needs to be chosen. The clean airfoil seems to match best with lower Ncrit - at least for the upper end of the drag bucket, the lower is well matched with higher Ncrit - while the gurney flap runs are better matched (in terms of maximum lift) at higher Ncrit overall. Even though this is a Gurney flap experiment, transition plays a major role in determining maximum lift and drag bucket.
# We go through the experiments using the list below,
# using hgv, the vector of gurney flap heights
# and using pchord, the position of the gurney flap
N = 180
AF = vft.repanel(vft.read_selig("S903RF.dat"),int(N),REFLIMS=[1.0,1.0,0.85,1.0],REFVAL=3)
def calc_drag(p,bl):
# Additional form drag from cavity
off = 0
drag = 0
for kf in range(len(p.foils)):
N = p.foils[kf].N
for ki in range(1,N):
signum = np.sign(p.foils[kf].S[ki]-bl[kf].ST)
drag += signum*0.5*(p.foils[kf].VD[off+ki]+p.foils[kf].VD[off+ki-1])*(p.cp[off+ki]-p.cp[off+ki-1])
off += N
return drag
# Virtual displacement
# 0 1 2 3 4 5 6
hgv = np.asarray( [0.0, 0.005, 0.01, 0.02, 0.02, 0.02, 0.01])
pchord = [1.0, 1, 1, 1, .95, .9, -1.0]
forced = np.asarray([0 , 0, 0, 0, 0, 0, 0])
pol = []
AOARANGE = []
AOARANGE.append(np.arange(0,-13,-0.5))
AOARANGE.append(np.arange(0.5,18,0.5))
# Go over experiment setup
for i in range(len(hgv)):
print("Running config %i of %i, "%(i+1,len(hgv)), end=" ")
hg = hgv[i]
pc = pchord[i]
# Gurney flap displacement
vd = 0*AF[0,:]
if pc>=1.0:
index_gf = N-1
index_gfp = N-2
elif pc<=-1.0:
index_gf = 0
index_gfp = 1
else:
for k in range(1,AF.shape[1]):
if AF[0,-k]<pc:
index_gf = N-k
index_gfp = N-k-1
break
vd[index_gf] = hg
# Solver Setup
RE = 1.0e6
# Not sure why the authors only state "Mach less than 0.2", with the chord length and RE of 1e6 it should be 0.1.
Mach = 0.1
# Ncrit = 6,7 is a better fit to the end of the drag bucket for the non-Gurney airfoil
# But for a gurney flap, it fits better with ncrit = 9 or 10
# Which indicates that we have a bit more complicated transition locations
if hg>0:
Ncrit = 9
else:
Ncrit = 7
s = vf.setup(Re=RE,Ma = Mach,Ncrit=Ncrit)
s.IterateWakes = False
s.Silent = True
alv = []
clv = []
cdv = []
cdvg = []
cdvg2 = []
cdex = []
cdex2 = []
unconverged = 0
for j in range(2): # aoa up and down
init = True
for alpha in AOARANGE[j]:
s.Alpha = alpha
if forced[i]:
s.Ncrit = 0
else:
s.Ncrit = Ncrit
if init:
# Set-up and initialize based on inviscid panel solution
(p,bl,x) = vf.init(AF,s)
xn = x.copy()
# Trigger transition a bit upstream of the GF
if hg>0:# and not forced[i]:
if pc>0:
xtrans = p.foils[0].X[0,index_gf]-0.02
vf.set_forced_transition(bl,p,[],[xtrans])
else:
xtrans = p.foils[0].X[0,index_gf]-0.02
vf.set_forced_transition(bl,p,[xtrans],[])
# Run 20 single iterations with slowly increasing flap
if hg>0:
s.Itermax = 1
for k in range(31):
[xn,flag,res,grad,_] = vf.iter(xn,bl,p,s,res,grad,vd*k/30)
s.Itermax = 100
init = False
res = None
grad = None
[xn,flag,res,grad,_] = vf.iter(xn,bl,p,s,res,grad,vd)
# Plot geometry
nres=np.sqrt(np.dot(res.T,res))
if flag>0:
# Add to end or start of polar list
if j == 1: pos = len(alv)
else: pos = 0
# Add pressure at gf. cp = 1-ue²/2
# cp_before: 1-0, cpafter: 1-ue²/2, cd: cp_before-cp_after
ue_gf = bl[0].bl_fl.nodes[index_gf].ue
ue_gfp = bl[0].bl_fl.nodes[index_gfp].ue
if abs(pc)<1:
ue_gfn = bl[0].bl_fl.nodes[index_gf+1].ue
cdvg.insert(pos,hg*(0.5*(ue_gf**2-ue_gfn**2)))
else:
ue_gfn = bl[0].bl_wk.nodes[0].ue
cdvg.insert(pos,hg*(0.5*(ue_gf**2)))
cdvg2.insert(pos,hg*(0.5*(ue_gfp**2)))
alv.insert(pos,alpha)
clv.insert(pos,p.CL)
cdv.insert(pos,bl[0].CD)
cdex2.insert(pos,calc_drag(p,bl))
cdex.insert(pos,abs(p.cp[index_gf-1]-p.cp[min(index_gf+1,N-1)])*hg)
else:
unconverged += 1
init = True
print("%u unconverged"%(unconverged))
pol.append({'CL': np.asarray(clv), 'CD': np.asarray(cdv), 'CDg':np.asarray(cdvg),'CDg2':np.asarray(cdvg2), 'hg': hg, 'al': np.asarray(alv),'cdex': np.asarray(cdex)})
Running config 1 of 7, 0 unconverged Running config 2 of 7, 0 unconverged Running config 3 of 7, 2 unconverged Running config 4 of 7, 3 unconverged Running config 5 of 7, 2 unconverged Running config 6 of 7, 1 unconverged Running config 7 of 7, 1 unconverged
matplotlib.rcParams['figure.figsize'] = [10, 7] # Make plots bigger than default
fig,ax = plt.subplots(1,2)
cmap = plt.get_cmap("tab10")
CDnam = 'CDg'
# Part one, influence of height
namesVF = ['VIIFLOW',None,None,None,\
None,None,\
'VIIFLOW forced',None,\
None]
linestyle_vf = ['-','-','-','-','-','-','-','-','-']
for k in range(4):
ax[0].plot(pol[k]['al'],pol[k]['CL'],'-',color=colors[k],label = namesVF[k])
ax[1].plot(pol[k]['CD']*1+1*(pol[k][CDnam]),pol[k]['CL'],color=colors[k])
#ax[1].plot(pol[k]['CD']*1+1*(pol[k]['cdex']),pol[k]['CL'],color=colors[k])
#ax[1].plot(pol[k]['CD']*1+1*(pol[k]['CDg2']),pol[k]['CL'],color='m')
namesEXP = ['Exp. Clean','Exp. GF 0.5%','Exp. GF 1%','Exp. GF 2%','Exp. GF 2% .95c',\
'Exp. GF 2% .9c','Exp. GF 1%\nUpper Surface']
for k in range(4):
Data = np.genfromtxt('S903GurneyCL.csv',skip_header=1,delimiter=",",usecols=[2*k,2*k+1])
ax[0].plot(Data.T[0],Data.T[1],color = colors[k],label = namesEXP[k],marker=marks[k],linestyle='None')
Data = np.genfromtxt('S903GurneyPolar.csv',skip_header=1,delimiter=",",usecols=[2*k,2*k+1])
ax[1].plot(Data.T[0],Data.T[1],color = colors[k],marker=marks[k],linestyle='None')
ax[0].legend(frameon=False,title='Influence of flap height')
ax[0].set_xlim(-15,20)
ax[1].set_xlim(0,0.04)
ax[1].set_ylim(-.75,1.6)
ax[0].set_ylim(-.75,1.6)
ax[0].set_xlabel('AOA (°)')
ax[0].set_ylabel('CL')
ax[1].set_xlabel('CD')
# Part two, influence of chord position
fig,ax = plt.subplots(1,2)
kcol = 0
for k in [0,3,4,5]:
ax[0].plot(pol[k]['al'],pol[k]['CL'],'-',color=colors[kcol],label = namesVF[k])
if k==0: useg=0
else: useg=1
ax[1].plot(pol[k]['CD']*1+useg*pol[k][CDnam],pol[k]['CL'],color=colors[kcol])
#ax[1].plot(pol[k]['CD']*1+useg*pol[k]['cdex'],pol[k]['CL'],color=colors[k])
kcol+=1
kcol = 0
for k in [0,3,4,5]:
Data = np.genfromtxt('S903GurneyCL.csv',skip_header=1,delimiter=",",usecols=[2*k,2*k+1])
ax[0].plot(Data.T[0],Data.T[1],':',color = colors[kcol],label = namesEXP[k],marker=marks[k])
Data = np.genfromtxt('S903GurneyPolar.csv',skip_header=1,delimiter=",",usecols=[2*k,2*k+1])
ax[1].plot(Data.T[0],Data.T[1],':',color = colors[kcol],marker=marks[k])
kcol+=1
ax[0].legend(frameon=False,title = 'Influence of flap position')
ax[0].set_xlim(-15,20)
ax[1].set_xlim(0,0.04)
ax[1].set_ylim(-.75,1.6)
ax[0].set_ylim(-.75,1.6)
ax[0].set_xlabel('AOA (°)')
ax[0].set_ylabel('CL')
ax[1].set_xlabel('CD')
# Part four, influence of upper/lower
fig,ax = plt.subplots(1,2)
kcol = 0
for k in [0,2,6]:
ax[0].plot(pol[k]['al'],pol[k]['CL'],'-',color=colors[kcol],label = namesVF[k])
useg=1
ax[1].plot(pol[k]['CD']*1+useg*pol[k][CDnam],pol[k]['CL'],color=colors[kcol])
kcol+=1
kcol = 0
for k in [0,2,6]:
Data = np.genfromtxt('S903GurneyCL.csv',skip_header=1,delimiter=",",usecols=[2*k,2*k+1])
ax[0].plot(Data.T[0],Data.T[1],':',color = colors[kcol],label = namesEXP[k],marker=marks[k])
Data = np.genfromtxt('S903GurneyPolar.csv',skip_header=1,delimiter=",",usecols=[2*k,2*k+1])
ax[1].plot(Data.T[0],Data.T[1],':',color = colors[kcol],marker=marks[k])
kcol+=1
ax[0].legend(frameon=False,title = 'Airfoil side')
ax[0].set_xlim(-15,20)
ax[1].set_xlim(0,0.04)
ax[1].set_ylim(-.85,1.6)
ax[0].set_ylim(-.85,1.6)
ax[0].set_xlabel('AOA (°)')
ax[0].set_ylabel('CL')
ax[1].set_xlabel('CD');
Pressure Distribution¶
In [7] the pressure coefficients over the airfoil surface is given for a lift coefficient of 0.7. The effect of the Gurney flap on the pressure is well predicted.
matplotlib.rcParams['figure.figsize'] = [10, 10] # Make plots bigger than default
N = 200
AF = vft.repanel(vft.read_selig("S903RF.dat"),int(N),REFLIMS=[1.0,1.0,0.85,1.0],REFVAL=3)
# Pressure comparison
hgv = np.asarray([0.0,0.02,0.02])
pchord = [1,1,.9]
alphas = [4.5,0.15+1.05,1.5+.6] # ->CL=0.7, read form above
lines = ['-','--',':']
shiftx = .4
shifty = 0
fig,ax = plt.subplots(2,1)
names = ['Clean VIIFLOW', 'GF VIIFLOW 2% 1c','GF VIIFLOW 2% .9c']
for k in range(len(hgv)):
hg = hgv[k]
pc = pchord[k]
alpha = 0*alphas[k]+5
vd = 0*AF[0,:]
if pc>=1.0:
index_gf = N-1
else:
for j in range(1,AF.shape[1]):
if AF[0,-j]<pc:
index_gf = N-j
break
vd[index_gf] = hg
s.Ncrit = 9
s.Alpha = alpha
s.CLTarget=.7
AFK = AF.copy()
AFK[1,:]-=k*.2
(p0,bl0,x0) = vf.init(AFK,s)
#x0[p.foils[0].N::p.foils[0].N+p.wakes[0].N-1]+=0.03
if hg>0 and True:
xtrans = p0.foils[0].X[0,index_gf]-0.001
vf.set_forced_transition(bl0,p0,[],[xtrans])
[x,flag,res,grad,_] = vf.iter(x0,bl0,p0,s,None,None,vd)
ax[0].plot(p0.foils[0].X[0,:]+k*shiftx,-p0.cp[0:p0.foils[0].N]+k*shifty,label=names[k],color = colors[k],linestyle=lines[k])
#print(flag,p0.CL) # Print to check the CL ~.7
vfp.plot_geometry(ax[1],p0,bl0);
names = ['Clean EXP', 'GF EXP 2% 1c','GF EXP 2% .9c']
for k in range(3):
Data = np.genfromtxt('S903GurneyCP.csv',skip_header=1,delimiter=",",usecols=[2*k,2*k+1])
ax[0].plot(Data.T[0]+k*shiftx,Data.T[1]+k*shifty,color = colors[k],label = names[k],marker=marks[k],linestyle="None")
ax[0].legend(frameon=False,ncol=2)
ax[0].set_xlabel('x/c, shifted by %g per case'%shiftx)
ax[0].set_ylabel('-CP');
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[2] Jeffrey, David, Xin Zhang, and David W. Hurst. Aerodynamics of Gurney flaps on a single-element high-lift wing. Journal of Aircraft 37.2 (2000): 295-301.
[3] Troolin, D. R., E. K. Longmire, and W. T. Lai. Time resolved PIV analysis of flow over a NACA 0015 airfoil with Gurney flap. Experiments in Fluids 41.2 (2006): 241-254.
[4] Coder, James George. CFD Investigation of Unsteady Rotorcraft Airfoil Aerodynamics. (2010).
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[6] Jeffrey, David, Xin Zhang, and David W. Hurst. Aerodynamics of Gurney flaps on a single-element high-lift wing. Journal of Aircraft 37.2 (2000): 295-301.
[7] Maughmer, Mark D., and Götz Bramesfeld. Experimental investigation of Gurney flaps. Journal of Aircraft 45.6 (2008): 2062-2067.
[8] Somers, Dan M. Effects of Airfoil Thickness and Maximum Lift Coefficient on Roughness Sensitivity: 1997--1998. No. NREL/SR-500-36336. National Renewable Energy Lab., Golden, CO (US), 2005.