surface pressure and viscous forces on inclined elliptic
TRANSCRIPT
Surface pressure and viscous forces on inclined elliptic cylindersin steady flow
SUBHANKAR SEN
Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines) Dhanbad,
Dhanbad 826 004, India
e-mail: [email protected]
MS received 29 January 2018; revised 25 September 2019; accepted 17 March 2020
Abstract. Surface pressure characteristics of elliptic cylinders of various thicknesses and orientations are
investigated in steady flow regime. A stabilized finite-element method has been used to discretize the conser-
vation equations of incompressible fluid flow in two dimensions. The Reynolds number, Re, is based on the
major axis of cylinder and free-stream speed. Results have been presented for Re� 40 and 0� � a� 90�, where ais the angle of attack. Cylinder aspect ratios AR considered are 0.2 (thin), 0.5 and 0.8 (thick). It is found that a
decrease in AR does not significantly alter the location of minimum surface pressure for a ¼ 90�, but the value ofminimum pressure decreases sharply, resulting in severe adverse pressure gradient. In contrast, for a ¼ 0�, thelocation travels towards the base and the minimum pressure increases, leading to delayed flow separation. In
general, the magnitude of forward stagnation pressure at low Re is smaller than the maximum pressure for
AR� 0:5. The maximum pressure occurs at the forward stagnation point as the Re and AR increase. However, in
most cases, the locations of forward stagnation and maximum pressure points differ even when the pressure
coefficients are very close to each other. The forward stagnation and maximum pressure coefficients of an
elliptic cylinder decrease monotonically with increasing a. The drag of a circular cylinder in most cases exceeds
the ones obtained for elliptic cylinders. With increasing AR, the drag increases approximately linearly for small
a, lift decreases approximately linearly and moment decreases non-linearly. For a thick cylinder, while the effect
of Re on lift and moment is insignificant, the drag shows a strong dependence. Roughly a ¼ 20� for Re ¼ 40
flow represents a critical angle of attack below which a cylinder of AR� 0:5 acts like a streamlined body and
above, like a bluff body.
Keywords. Stabilized finite-element; elliptic cylinder; angle of attack; pressure coefficient; bluffness.
1. Introduction
The flow around elliptic cylinders is associated with physics
that are displayed by both streamlined as well as bluff bodies.
A wide range of geometries, i.e. flat plate to circle, can be
simulated by altering the ratio of length of minor to major
axes or aspect ratio AR of an elliptic cylinder. The angle of
attack a of an elliptic cylinder is formed between the
incoming stream and its major axis (see figure 1). Symmetric
configuration of an elliptic cylinder corresponds to a ¼ 0� or90�; intermediate values of a lead to asymmetric configura-
tion. Compared with its circular (AR ¼ 1) and square coun-
terparts, the analysis of flow past elliptic cylinders has
received much lesser attention. Elliptic tubes find wide
applications in heat exchangers owing to much lesser
pumping requirements as compared with circular tubes [1].
The early investigations concerning flow past elliptic
cylinders were confined mostly to semi-analytical or
numerical treatment of the conservation equations at very
low Reynolds number, Re. For a ¼ 0� and steady flow at
Re ¼ 40, Dennis and Chang [3] predicted the drag on
elliptic cylinders of varying AR. By approximately solving
Oseen’s linearized equations in two dimensions, Yano and
Kieda [27] presented limited results highlighting the
dependence of drag coefficient Cd (for AR ¼ 0:1, 0.5 and 1)and lift coefficient Cl (for AR ¼ 0:1 and 0.5) on a. Theresults were presented for Re ¼ 0:1 and 1. With aincreasing from 0� to 90�, they observed monotonic rise of
Cd and a non-monotonic variation of Cl. By solving the
Stokes equations via finite-element method, Sugihara-Seki
[25] explored, for Re � 0, the motion of a freely suspended
elliptic cylinder in a narrow channel. Masliyah and Epstein
[15] employed finite-difference method with relaxation to
discretize the Navier–Stokes equations of motion cast in the
streamfunction–vorticity (w� x) form. For two-dimen-
sional steady flow past symmetrically oriented elliptic
cylinders of AR ¼ 0:2� 1, they presented results for
Re� 90. By employing finite-difference on the w� xequations, Lugt and Haussling [14] investigated the steady
Sådhanå (2020) 45:172 � Indian Academy of Sciences
https://doi.org/10.1007/s12046-020-01397-z Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)
and unsteady flows around thin elliptic cylinders of AR =
0.1 and 0.2 at 45� incidence. They presented detailed resultsfor fluid loading and streamline patterns at Re ¼ 15, 30 and
200. In order to explore the steady flow past a thin elliptic
cylinder of AR ¼ 0:2 at incidence, D’Alessio and Dennis
[2] used a semi-analytical method for the w� x equations.
They reported results for Re ¼ 5 and 20. Using a semi-
analytical method, Dennis and Young [4] presented, in the
steady flow regime, detailed results for the aerodynamic
coefficients of elliptic cylinders at incidence besides
exploring the separation topology. They considered thin
elliptic cylinders of AR ¼ 0:1 and 0.2. Faruquee etal [5]
numerically investigated the effects of AR on characteris-
tics of steady flow at Re ¼ 40. For elliptic cylinders of
AR ¼ 0:3� 1 and major axis parallel to the incoming
stream (a ¼ 0�), this study provides a detailed analysis of
the relationship between Cd and its pressure and viscous
components, Cdp and Cdv, respectively, with AR. For steady
unbounded flow of power-law fluids around symmetric
elliptic cylinders of AR ¼ 0:2� 1, Sivakumar etal [21]
reported extensive numerical results for fluid forces and
surface pressure besides exploring the separation phe-
nomenon. The Re range considered was 0.01-40. Sen etal
[20] presented detailed numerical results concerning steady
separation of laminar boundary layer for symmetrically as
well as asymmetrically oriented elliptic cylinders of various
AR. They proposed separation topologies for both config-
urations of the cylinder. For the inclined cylinders, signif-
icant deviation in separation topology was found relative to
those proposed earlier by Smith [22] and Dennis and Young
[4]. For elliptic cylinders of AR ¼ 0:2, 0.5 and 0.8, Sen and
Mittal [18] numerically investigated, at Re ¼ 200, the
evolution of far wake vortical structures as a function of a.Very recently, Sourav etal [24] numerically studied the
response of a freely vibrating thick elliptic cylinder of
AR ¼ 109at low Reynolds numbers.
A distinguishing feature of the surface pressure, Cp,
distribution of thin (AR of O(0.2) or smaller) elliptic
cylinders as compared with those of a circular cylinder is
that the magnitude as well as location of the maximum
pressure, Cpmax, in the former deviates from those of the
forward stagnation pressure, Cpfs, at low Re, such as 5 or
smaller. Little information is available in the literature on
this interesting phenomenon. For instance, the presence of a
maximum in Cp of an elliptic cylinder at locations other
than the leading edge or forward stagnation point was
earlier reported by Masliyah and Epstein [15] solely for the
symmetric configuration of a ¼ 90�. However, no referencewas drawn by them to the forward stagnation and maximum
pressure coefficients. At low Re, the departure of Cpmax
from Cpfs in magnitude as well as location is also evident
from figures 17b and c of Yoon etal [28] for steady flow
past a square cylinder at 0� and 45� incidences, respec-
tively. They did not report or explore this phenomenon. In
general, for steady flow around inclined elliptic cylinders of
various thicknesses, a detailed analysis of surface pressure
distribution as well as a comprehensive data set for the fluid
forces is unavailable in the literature. These research gaps
motivate the current study. In this computational work, the
features of surface pressure distribution on symmetrically
as well as asymmetrically oriented elliptic cylinders in an
unbounded medium are studied. The departure of the
maximum pressure from the forward stagnation pressure is
investigated and a possible explanation is provided. To this
end, steady laminar flow past elliptic cylinders of AR ¼ 0:2(thin), 0.5 and 0.8 (thick) at incidence have been considered
for Re� 40. For each AR, the angle of attack is varied from
0� to 90� in steps of 15�. Very limited results have also
been presented for the circular cylinder. A stabilized finite-
element solver that uses the same order of bilinear inter-
polation for fluid velocity, u ¼ ðu; vÞ, and pressure, p, has
been employed in two dimensions. The discretized lin-
earized algebraic equations have been generated using the
simultaneous or full coupling method, resulting in a single
global matrix equation system for the dependent primitive
variables, i.e. velocity and pressure.
The remaining of this article is organized in the fol-
lowing manner. In section 2, the governing differential
equations for incompressible fluid flow are reviewed. The
finite-element formulation involving SUPG (streamline-
upwind/Petrov-Galerkin) and PSPG (pressure-stabilizing/
Petrov-Galerkin) stabilization is discussed in section 3. The
definition of the problem and finite-element mesh are
depicted in sections 4 and 5, respectively. Validation of the
formulation along with mesh convergence is discussed in
section 6. The main results are presented and discussed in
section 7. In section 8, a few concluding remarks are made.
2. The governing equations
Let X � R2 represents the spatial domain and its boundary
C be piecewise smooth. The spatial coordinates are denoted
by the vector x = (x, y). In strong form, the conservation
,
,
σσ
αxx
yx
vu
u
u
v
v
x
y
= U = 0
= U
= U
= 0
= 0
= 0
= 0
u = 0 v,
baθ
80 a 120a
100a
= 0
base point
Figure 1. Problem set-up for steady unbounded flow around a
stationary elliptic cylinder inclined at an angle a to the free-
stream. The base point is marked in the figure. For this study,
AR ¼ 0:2; 0:5 and 0.8. This figure is not drawn to scale.
172 Page 2 of 18 Sådhanå (2020) 45:172
equations governing the steady flow of an incompressible
fluid of density, q, are
qðu � $u� fÞ � $ � r ¼ 0 on X; ð1Þ
$ � u ¼ 0 on X: ð2ÞEquations (1) and (2) are cast in vector or coordinate-free
form. The choice of stress-divergence form of the Navier–
Stokes equations (Eq. (1)) is driven by the fact that the
resultant weak form contains the surface traction vector, h(defined in Eq. (4)) as the flux or natural boundary condi-
tion term (the right hand side term of Eqs. (5) and (6)). In
these equations, f and r denote the body force per unit
volume and the Cauchy stress tensor at a point, respec-
tively. The constitutive relation for stress in terms of its
inviscid or isotropic and viscous or deviatoric contributions
reads as
r ¼ �pIþ 2leðuÞ; where eðuÞ ¼ 1
2ðð$uÞ þ ð$uÞTÞe:
ð3ÞIn Eq. (3), I, l and e stand for the identity tensor, dynamic
viscosity of the fluid and strain rate tensor, respectively.
The essential or Dirichlet and natural or Neumann-type
boundary conditions are represented as
u ¼ g on Cg; n � r ¼ h on Ch; ð4Þrespectively. Here, Cg and Ch are complementary sub-
sets of C and n is its local unit normal vector. In the
present work, the free-stream or towing tank boundary
condition is used on the lateral boundaries of the
domain (figure 1). This requires prescribed free-stream
speed condition on the upstream as well as lateral
boundaries. The no-slip condition on velocity is applied
along the surface of the cylinder. At the exit/down-
stream boundary, a Neumann condition for velocity is
specified that amounts to traction-free condition. The
boundary conditions are illustrated in figure 1. The
choice of boundary conditions as well as blockage
(defined in section 4) in the present work is represen-
tative of unbounded flow [17], [19].
3. Stabilized finite-element formulation
The spatial domain X is discretized into non-overlapping
quadrilateral elements or sub-domains Xe, e ¼ 1; 2; :::; nelwhere nel is the total number of elements. Let Sh
uand Sh
p
denote the finite-dimensional trial function spaces for u and
p, respectively. The respective weighting function spaces
are denoted by Vhuand Vh
p. The stabilized finite-element
formulation of the conservation equations (1) and (2)
combined (as in a coupled formulation) is written as
follows: find uh 2 Shuand ph 2 Sh
p such that 8wh 2 Vhu,
qh 2 VhpZ
Xwh � q uh � $uh � f
� �dX
þZXeðwhÞ : rðph; uhÞdX
þXnele¼1
ZXe
1
qsSUPGqu
h � $wh þ sPSPG$qh
� �:
q uh � $uh � f� �� $ � rðph; uhÞ� �
dXe
þZXqh$ � uhdX
þXnele¼1
ZXe
d$ � whq$ � uhdXe ¼ICh
wh � hhdC:
ð5Þ
In the combined weak or variational formulation of
Eqs. (1) and (2) presented by Eq. (5), the first, second
and fourth terms in the left hand side along with the right
hand side construct the Galerkin statement of the prob-
lem. The first series of element level integrals are the
SUPG and PSPG stabilization terms added to the varia-
tional formulations of the momentum and continuity
equations, respectively. The SUPG term provides stability
against spurious oscillations in the velocity field while
the PSPG term suppresses the spurious modes of pressure
that might appear owing to the use of the same order of
interpolation for velocity and pressure. The inclusion of
PSPG term ensures non-zero coefficient of pressure in the
discretized continuity equation and hence, introduces
definiteness in the matrix equation system. The second
series of element level integrals enhance numerical sta-
bility of the formulation at high Re. Equation (5) can be
decomposed in its respective momentum and continuity
components as
ZXwh � q uh � $uh � f
� �dX
þZXeðwhÞ : rðph; uhÞdX
þXnele¼1
ZXe
1
qsSUPGqu
h � $wh� �
:
q uh � $uh � f� �� $ � rðph; uhÞ� �
dXe
þXnele¼1
ZXe
d$ � whq$ � uhdXe ¼ICh
wh � hhdC:
ð6Þ
ZXqh$ � uhdXþ
Xnele¼1
ZXe
1
qsPSPG$q
h� �
:
q uh � $uh � f� �� $ � rðph; uhÞ� �
dXe ¼ 0:
ð7Þ
In the present work, f ¼ 0. More details of the finite-
element formulation are available in Tezduyar etal
[26].
Sådhanå (2020) 45:172 Page 3 of 18 172
4. The problem definition
An elliptic cylinder with major axis of length ‘a’ and minor
axis of length ‘b’ is placed in a computational domain
whose exterior boundary is chosen to be a rectangle (see
figure 1). Aspect ratio of the cylinder is computed as
AR ¼ ba. The origin of the Cartesian coordinate is located at
the centre of the fixed cylinder. The positive x axis extends
along the downstream. Relative to the negative x axis, the
angle of attack is measured clockwise whereas the cir-
cumferential angle h is measured counterclockwise. For all
a, the trailing tip represents the base point of the cylinder
(highlighted in the figure). The streamwise distance of the
upstream and downstream boundaries of the domain mea-
sured from the cylinder centre are fixed to 80a and 120a,
respectively. The lateral boundaries are equidistant from
the centre of the cylinder; the domain width or vertical
distance between these boundaries is fixed to 100a for all
the computations. This arrangement results in a blockage of
0.01 for a ¼ 90� and it is lower for a\90�. Blockage is theratio of cross-stream projection of the cylinder to the
domain width. The Reynolds number is based on the major
axis and free-stream speed, U, i.e. Re ¼ Uam where m is the
kinematic viscosity of the fluid. The moment exerted by the
fluid on the cylinder is calculated at the centre of the
cylinder. The clockwise moment is considered positive.
The following definitions are used for the fluid forces
(Cd;Cl) and moment (Cm) coefficients:
Cd ¼ 112qU2a
ZCcyl
hxdC; ð8Þ
Cl ¼ 112qU2a
ZCcyl
hydC; ð9Þ
Cm ¼ 112qU2a2
ZCcyl
r� hdC: ð10Þ
Here, hx and hy denote the streamwise and cross-stream
components, respectively, of the surface traction vector,
Ccyl represents the cylinder boundary and r is the radius
vector of any arbitrary point located on the cylinder surface
measured from the centre of cylinder.
5. Finite-element mesh
The computational domain is discretized using a non-uniform,
multiblock and structured mesh consisting of bilinear quadri-
lateral elements. Irrespective of AR and a, the number of nodes
and number of elements for each mesh are fixed to 120626 and
119768, respectively. Figure 2a shows a representative finite-
element mesh for AR ¼ 0:5 and a ¼ 45�. A close-up of the
mesh near the cylinder is shown in figure 2b. Five contiguous
mesh blocks, i.e. a central square block accommodating the
cylinder and its four neighbour rectangular blocks directed
along the left, right, top and bottom, constitute the mesh. The
central block is non-Cartesian; it is composed of two families of
orthogonal grid lines, i.e. radial and circumferential. Each cir-
cumferential grid line in the central block contains 464 nodes.
The normalized radial thickness of the first layer of elements
lying on the cylinder surface is 0.0005. The finite-element
meshesused in thiswork are identical to thoseused inour earlier
work (see Sen etal [20]).
(b)(a)
x
y
Figure 2. Steady flow past an elliptic cylinder of AR ¼ 0:5: (a) finite-element mesh corresponding to a ¼ 45� and (b) its close-up near
the cylinder. The mesh consists of 120626 nodes and 119768 bilinear quadrilateral elements. In (b), the origin of the coordinate system is
marked by a � symbol.
172 Page 4 of 18 Sådhanå (2020) 45:172
6. Validation and convergence of predicted results
6.1 Comparison to the earlier studies
For steady flow past elliptic cylinders, extensive validation
of fluid forces by us has been reported in Sen etal [20] and
Sourav and Sen [23].
6.2 Mesh convergence
The mesh convergence of the predicted results has been
established in one of our earlier works. Table 5 of Sen etal
[20] provides a detailed discussion on mesh convergence of
results.
7. Results
The aerodynamic behaviour of elliptic cylinders at inci-
dence is studied numerically for Re� 40 in the steady flow
regime. The aspect ratios of elliptic cylinders are 0.2, 0.5
and 0.8, while the angle of attack varies from 0� to 90�. Theelement level flow matrix and vector entries have been
computed by employing the 2� 2 points Gauss–Legendre
quadrature rule. The linearized asymmetric algebraic
equation system of flow variables has been solved by a
matrix-free GMRES or Generalized Minimal RESidual
method proposed by Saad and Schultz [16]. A diagonal pre-
conditioner has been used to accelerate the convergence of
non-linear iterations.
(a) (b)
(c)
Fig. 3. Steady unbounded flow past elliptic cylinders at Re ¼ 40: the surface vorticity distribution for cylinders of AR ¼ 0:2, 0.5 and 0.8for a ¼ (a) 0�, (b) 45� and (c) 90�. In (b), the location of forward stagnation point is highlighted for AR ¼ 0:2.
Sådhanå (2020) 45:172 Page 5 of 18 172
In an early numerical exploration aiming the determi-
nation of the critical Reynolds number, Rec, marking the
onset of vortex-shedding or transition to the unsteady
regime of flow, Jackson [10] considered objects of several
shapes, including circular and elliptic cylinders of varying
AR. Table 2 of Jackson [10] lists the values of Rec and
corresponding Strouhal number, Stc, for a flat plate, a cir-
cular cylinder and elliptic cylinders of various AR. From
this table, the values of Rec for AR ¼ 0:1 and 0.3 elliptic
cylinders are 29.680 and 32.765, respectively. These
numerical values underscore that the flow past the AR ¼0:2 cylinder is unsteady at the maximum Re of 40 consid-
ered in the current study. For AR ¼ 0:5, table 5 of Jackson
[10] summarizes the values of Rec and Stc with a. For
a ¼ 90�, Rec ¼ 35:704 and the values of Rec and Stcincrease monotonically as a continues to decrease to 0�.Johnson etal [11] performed two-dimensional spectral-
element computations and investigated the flow past elliptic
cylinders of AR ¼ 0:01� 1 (or major axis normal to flow)
over Re ¼ 30� 200. Figure 4 of their paper illustrating
vorticity field for AR ¼ 0:5 at Re ¼ 40 ascertains that
vortex-shedding is absent and the flow is steady. As evident
from figure 14 of Johnson etal [11], the flow for AR ¼ 0:25turns unsteady below Re ¼ 40. This figure also hints
towards a monotonic decay of Rec with reducing AR. In
view of this variation of critical Re with AR, the flow past
the AR ¼ 0:2 cylinder at Re ¼ 40 is expected to be essen-
tially unsteady. By artificially preventing the onset of vor-
tex-shedding or stabilizing the wake, a flow can, however,
remain in the steady regime at high Re (see Grove etal [8],
Fornberg, [6, 7], Sen etal [19], etc.). In the current work, we
compute the Re ¼ 40 flow past the AR ¼ 0:2 cylinder as a
steady flow. Dennis and Young [4] and Sivakumar etal [21]
also performed similar steady state computations.
7.1 Surface vorticity
Figure 3 illustrates, at Re ¼ 40, the distribution of nor-
malized surface vorticity, xDU, for symmetrically and
asymmetrically oriented elliptic cylinders of AR = 0.2, 0.5
and 0.8. It is observed that the symmetric configuration of a
cylinder leads to anti-symmetric vorticity distribution about
the base point. In this work, the aerodynamic forces have
been computed from the surface pressure and viscous
stresses along the surface. The anti-symmetry of vorticity
ensures absence of lift force for the symmetric cylinders,
i.e. a ¼ 0� and 90�. For asymmetric configurations, x dis-
plays asymmetry about the base and hence, lift force exists.
When the laminar boundary layer is attached to the cylinder
surface, the vorticity curves along the cylinder surface are
characterized by the presence of two zero-vorticity or zero-
shear stress singular points (AR = 0.2 case in figure 3a for
vorticity), i.e. the forward and rear stagnation points [13].
The appearance of two more zero-vorticity points, corre-
sponding to the separation and attachment points, implies
separated flow (AR = 0.5 and 0.8 cases in figure 3a and b,
c). The separation of laminar boundary layer from elliptic
cylinders has been discussed in Sen etal [20].
For a representative Reynolds number of 20, the classical
closed wake profiles for AR ¼ 0:2, 0.5 and 0.8 elliptic
cylinders with a ¼ 90� orientation are plotted together in
figure 4. For comparison, also shown is the wake of the
AR ¼ 1 cylinder. The largest size of wake is associated
with AR ¼ 0:2 whereas the shortest and narrowest wake
corresponds to AR ¼ 1. In particular, the length and width
of the wake decrease with increasing AR. It is discussed
later that when a ¼ 90�, the AR ¼ 0:2 cylinder is the most
bluff and the AR ¼ 0:8 cylinder is the least bluff. As shown
by Kumar etal [12], the topology of a symmetric closed
wake satisfies the kinematic constraint of Hunt etal [9]
discussed in section 7.2.
7.2 Features of surface pressure
Pressure distribution along the surface of elliptic cylin-
ders at Re = 40 is presented in figure 5 for a ¼ 0�, 45� and90�. The pressure coefficient is defined as Cp ¼ p�p0
12qU2 where
p0 is the free-stream pressure. In this work, the free-stream
pressure is assigned a zero value. Sharp spikes character-
ized by large negative value of pressure are seen in the
Cp � h curves for a ¼ 45� and 90�. At the location of
minimum pressure, Cpmin, the pressure gradient changes
sign from negative to positive or becomes adverse (also see
Faruquee etal [5]). Between the points of minimum pres-
sure, favourable pressure gradient persists along the front
surface of the cylinders and adverse pressure gradient along
the aft. The Cp � h distribution around the surface of
symmetric cylinders (figure 5a, c) is characterized by
identical pressure variation along the lower (0� � h� 180�)and upper (180� � h� 360�) surfaces. Also, in all such
cases, the forward stagnation point is located at h ¼ 0� (or
360�) and the corresponding forward stagnation pressure
represents (for Re ¼ 40) the maximum pressure on the
cylinder. The forward stagnation point is a point of zero-
Figure 4. Steady separated flow past elliptic cylinders with
major axis perpendicular to the flow in an unbounded medium:
comparison of the wake size via contours of w for AR ¼ 0:2, 0.5,0.8 and 1 at Re ¼ 20.
172 Page 6 of 18 Sådhanå (2020) 45:172
vorticity such that vorticity in its neighbourhood changes
sign from negative to positive (see figure 3b for AR ¼ 0:2at Re ¼ 40). Thus, location of the forward stagnation point
is determined from the surface vorticity distribution.
Details on identification of location of forward stagnation
point from surface vorticity are available in Sen etal [20].
In figures 5 and 7 of Sen etal [20], the point P denotes the
forward stagnation point. For a ¼ 0� (figure 5a), the
minimum pressure for the AR = 0.8 cylinder is obtained for
h slightly greater than 90� or smaller than 270�. However,as AR decreases, the minimum pressure points move
towards the base. Also, the value of Cpmin increases, i.e.
Cpmin becomes less negative. This is consistent with the
observations of Faruquee etal [5]. For AR ¼ 0:2, the min-
imum is achieved at the base point. The flow remains fully
attached under strong favourable pressure gradient and the
streamlines closely follow the body contour (see figure 6c).
Figure 6a shows the surface pressure and its gradient,oCp
os ,
while a close-up of the same is presented in figure 6b. The
gradient is measured along the cylinder contour (the
direction is denoted by s) between the stagnation points.
Existence of a very weak adverse pressure gradient in the
vicinity of the base point is apparent from figure 6b. When
a ¼ 90� (figure 5c), Cpmin for AR ¼ 0:2 is obtained for hslightly greater than 90� or smaller than 270�. The locationof Cpmin appears practically uninfluenced by AR (contrary
to the case of a ¼ 0�), while its value declines sharply with
decreasing AR. The severity of adverse pressure gradient
therefore increases with decreasing thickness and leads to
-2.5
-2.0
-1.0
0.0
1.0
2.0
2.5
0 90 180 270 360
Cp
θ (deg.)
AR = 0.2 0.5 0.8
-2.5
-2.0
-1.0
0.0
1.0
2.0
2.5
0 90 180 270 360
Cp
θ (deg.)
AR = 0.2 0.5 0.8
-2.5
-2.0
-1.0
0.0
1.0
2.0
2.5
0 90 180 270 360
Cp
θ (deg.)
AR = 0.2 0.5 0.8
(a) (b)
(c)
90
45a = 0
Figure 5. Steady unbounded flow at Re ¼ 40 past elliptic cylinders of AR ¼ 0:2, 0.5 and 0.8: distribution of surface pressure for a = (a)
0�, (b) 45� and (c) 90�.
Sådhanå (2020) 45:172 Page 7 of 18 172
earlier (at lower Re) separation of the laminar boundary
layer. For AR ¼ 0:2, figure 7 shows strong adverse pressure
gradient near the cylinder shoulders (h � 90� or 270�) andthe wake comprising an attached separation bubble.
Asymmetry about the h ¼ 180� location in the Cp � hcurves is introduced when the cylinders are oriented
asymmetrically (figure 5b). For each AR the minimum
pressure on the upper rear surface is smaller than the
minimum pressure on the lower front surface, implying
stronger adverse pressure gradient on the upper rear sur-
face. This effect is more pronounced with reducing thick-
ness. The magnitude of minimum pressure of a symmetric
elliptic cylinder decreases with increasing Re (see figure 8a
and c for AR ¼ 0:2). For the range of Re considered, it is
also noted that the location of Cpmin for symmetric con-
figurations of the AR ¼ 0:2 cylinder, i.e. a ¼ 0� and 90�, ispractically insensitive to Re. For a ¼ 0�, the minimum is
attained at the base point (figure 8a) and for a ¼ 90�, at theshoulders (figure 8c and seventh column of table 1). As
evident from figure 8b for a ¼ 45�, the location of Cpmin for
asymmetric orientations, in contrast, depends on Re. The
location of Cpmin for symmetric cylinders of AR ¼ 0:5 and
0.8 moves upstream with increasing Re (not shown).As
stated in section 7.1, Sen etal [20] studied the steady sep-
aration of laminar boundary layer from inclined elliptic
cylinders of AR ¼ 0:2, 0.5 and 0.8. They explored the
separation topology for symmetric as well as asymmetric
configurations of the cylinders. As outlined in Kumar etal
[12], the wake structure for symmetric configurations
relates to the classical closed wake comprising a pair of
counter-rotating eddies (e.g. figures 4, 6c, 7c and 16d of
the present paper) whereas for asymmetric orientations, the
wake is open and consists of an attached (to the rear of the
cylinder) and a detached vortex (figure 9a of the present
paper for AR ¼ 0:8; a ¼ 45� and Re ¼ 30). The open wake
for asymmetric cylinders represents an instance of alleyway
flow. Streamline patterns for elliptic cylinders with sym-
metric/asymmetric configurations were presented in fig-
ures 3 and 4 of Sen etal [20]; schematics of the separation
topology for both types of orientations were shown in
-30
-25
-20
-15
-10
-5
0
5
0 90 180 270 360
Cp,
dCp/
ds
θ (deg.)
CpdCp/ds
-0.8
-0.6
-0.4
-0.2
0
0.2
0 90 180 270 360
s
s
(b)(a) (c)
U
Figure 6. Steady unbounded flow past an elliptic cylinder of AR ¼ 0:2 for a ¼ 0� at Re ¼ 40: (a) distribution of surface pressure andoCp
os along the cylinder contour, (b) close-up of (a) and (c) streamlines indicating attached flow.
(b)(a) (c)
-35-30-25-20-15-10-5 0 5
10 15
0 90 180 270 360
Cp,
dCp/
ds
θ (deg.)
CpdCp/ds
-2
-1
0
1
2
0 90 180 270 360
Figure 7. Steady unbounded flow past an elliptic cylinder of AR ¼ 0:2 for a ¼ 90� at Re ¼ 40: (a) distribution of surface pressure and
its gradient along the cylinder contour, (b) close-up of (a) and (c) streamlines indicating separated flow with a closed attached wake.
172 Page 8 of 18 Sådhanå (2020) 45:172
figure 5. The cylinder contour, known as the surface
streamline, is conventionally treated as the zero (reference)
streamline. Thus, the value of the streamfunction is zero for
this streamline. All streamlines emanating from or
terminating to the reference streamline are zero streamlines
and the points of intersection of streamlines are known as
critical points, where the vorticity and shear stress disap-
pear. The critical points are broadly classified as nodes and
-3
0
3
6
0 90 180 270 360
Cp
θ (deg.)
Re = 52040
-4
-3
-2
-1
0
1
2
3
4
5
0 90 180 270 360
Cp
θ (deg.)
-5
-4
-3
-2
-1
0
1
2
0 90 180 270 360
Cp
θ (deg.)
(a)
(c)
(b)
45
90
0α =
Figure 8. Distribution of surface pressure for steady unbounded flow past an elliptic cylinder of AR ¼ 0:2 for Re ¼ 5; 20 and 40 and
a ¼ (a) 0�, (b) 45� and (c) 90�.
Table 1. Steady unbounded flow past an elliptic cylinder of AR ¼ 0:2; a ¼ 90� for Re ¼ 1; 6; 7 and 40: summary of the values of
maximum and minimum Cp along with the h location. Also listed in the last column are the values of Cp at the shoulder. For Re� 6,
Cpfs\Cpmax and for Re 7, Cpfs ¼ Cpmax.
AR Re Cp0 Cpmax h for Cpmax Cpmin h for Cpmin Cp at h ¼ 90�
0.2 1 3.7115 7.5124 86.90� –9.9810 92.33� –2.8959
0.2 6 1.5624 1.6256 86.12� –4.1923 91.55� –2.5845
0.2 7 1.4900 1.4900 0� –3.9376 91.55� –2.5489
0.2 40 1.1024 1.1024 0� –1.8616 90.78� –1.7155
Sådhanå (2020) 45:172 Page 9 of 18 172
saddles. The zero streamlines along with the critical or
singular points constitute the separation topology. Regard-
ing kinematic stability of a separation topology, Hunt etal
[9] presented the following constraint to be satisfied:
XN þ 1
2
XN 0
� ��
XSþ 1
2
XS0
� �¼ 1� m: ð11Þ
The quantities N and S appearing in Eq. (11) signify the
four-way nodes and four-way saddles, respectively. The
three-way nodes and three-way saddles are, respectively,
denoted by symbols N 0 and S0. The connectivity of the flow
section is denoted by m; in the context of flow past a
solitary obstacle, m ¼ 2. For the AR ¼ 0:8 cylinder with
a ¼ 45�, figure 9a depicts, via streamlines, the alleyway
flow at Re ¼ 30. The corresponding separation topology is
shown schematically in figure 9b, where A and S, respec-
tively, stand for attachment and separation points that
appear alternately along the fluid–solid interface. They are
no-slip type three-way (or half) saddle points. The eddy
centres, denoted by N1 and N2, are four-way nodes. The
sole four-way saddle point, denoted by M, exists in the fluid
medium. Point M is the asymmetric equivalent to the wake
stagnation point that exists in a symmetric, closed wake
(marks the closure of the wake). With this identification of
critical points,P
S ¼ 1 (point M),P
N ¼ 2 (points N1 and
N2),P
N 0 ¼ 0 andP
S0 ¼ 4 (points A1, A2, S1 and S2).
The left hand side of Eq. (11) reduces to
2þ 0� 1� 42¼ �1. The right hand side, i.e. 1� m, also
yields the same value of �1. Thus, the separation topology
of alleyway flow proposed by Sen etal [20] is stable.
For Re ¼ 5� 40, figure 10 depicts the relationship
between the forward stagnation pressure coefficient, Cp0,
and Re. Irrespective of AR and a, the pressure coefficient
drops with Re. For a ¼ 90�, Cp0 appears to be the least
sensitive to AR. The relationship between base suction,
�Cpb, and Re is illustrated in figure 11a, b and c for
a ¼ 0�, 45� and 90�, respectively. For fixed AR and a, thebase suction of a cylinder decreases with increasing Re. In
general, �Cpb for a fixed Re decreases with increasing AR.
The decay of base suction signifies recovery of pressure at
the base of the cylinder. This eventually leads to lower
pressure drag.
Figures 12a, b and 13a–c, respectively, demonstrate that
Cpmax for low Re does not occur at the forward stagnation
point of symmetric (here, a ¼ 90�) and asymmetric cylin-
ders of AR� 0:5. As revealed by figure 12a, the maximum
surface pressure on a thin cylinder with a ¼ 90� at low Re,
i.e. Re� 6, occurs near its shoulders instead of the forward
stagnation point. As Re is increased, the maximum pressure
point travels upstream along the cylinder surface (or w ¼ 0
streamline) and coincides with the forward stagnation point
for Re[ 6. The departure of Cpfs from Cpmax is also dis-
played by the AR ¼ 0:5 cylinder at Re ¼ 1 (see the inset in
figure 12b), where the maximum pressure points are loca-
ted upstream of the shoulders. With increasing cylinder
thickness (AR ¼ 0:8, for instance), the difference between
Cpfs and Cpmax diminishes (inset of figure 12c) and the
maximum pressure point merges with the forward stagna-
tion point for all Re. As mentioned in section 1, the pres-
ence of a maximum in Cp at low Re was earlier reported by
Masliyah and Epstein [15] for AR ¼ 0:2 and a ¼ 90�.Though it is not explicitly pointed out, similar observations
can be made from the pressure distribution curves presented
by Sivakumar etal [21] for AR = 0.2 and a ¼ 90� at very
low Re (= 0.01). For Re ¼ 5 and 40, figure 13a–c demon-
strates that both Cpfs and Cpmax decrease monotonically
with increasing a. The (Cpmax � Cpfs) difference decreases
with increasing Re as well as AR, and approaches zero for
the thick cylinder. It is also noted that both Cpfs and Cpmax
decrease with increasing Re. The dependence of h locations
of the forward stagnation (hfs) and maximum (hmax) pres-sure points on a is depicted via figure 13d–f. For
0� � a� 90�, both hfs and hmax, in general, display non-
monotonic variation with a. A departure from the non-
monotonic hmax � a trend is found to occur for a thin
cylinder at Re ¼ 5 (figure 13d), which shows linear
decrease of hmax with a. Interestingly, hfs and hmax differ inmost cases (such that hfs [ hmax or the maximum pressure
point resides above the forward stagnation point) even
when the values of Cpfs and Cpmax are very close to each
MN
NA1
S1
A
S
1
22
2
(a) (b)
ψ = 0
Figure 9. Steady separated flow past an inclined elliptic cylinder of AR ¼ 0:8 with a ¼ 45�: (a) streamlines depicting the asymmetric
wake at Re ¼ 30 and (b) schematic of corresponding separation topology illustrating the critical points.
172 Page 10 of 18 Sådhanå (2020) 45:172
other. For an asymmetric cylinder, the second row of fig-
ure 13 indicates that the difference between hfs and hmaxdecreases with increasing Re as well as AR.
This mismatch in magnitude as well as location of for-
ward stagnation pressure and maximum pressure at low Re
appears to be governed by a pair of factors, i.e. the Rey-
nolds number and the streamwise extent of the body. A low
value of Re and abrupt change of body contour appear to
facilitate the occurrence of this phenomenon. Following the
discussion in section 1 it is obvious that the shoulders of
diamond, thin elliptic cylinders and front corners of a
square cylinder are ideal locations where Cp at low Re turns
the maximum and hence, exceeds Cpfs. It may be noted that
the mismatch is absent for a ¼ 0� even at an Re as small as
0.01 (see the first row of figure 12 of Sivakumar etal [21]
corresponding to the value of the power-law index being
unity). For this configuration, the streamwise extent of the
cylinder is appreciably high. At low Re, the mode of
diffusion transport dominates and the pressure scales with
U. As Re is increased, transport of vorticity by convection
becomes appreciable. In addition, the order of magnitude of
pressure equals the square of U. For AR ¼ 0:2 and 0.8
cylinders at zero incidence, figure 14a and b, respectively,
depicts the relationship of logarithm of Cp0 and Cpb with
logarithm of Re. Since the pressure coefficients decay with
Re or follow an inverse relationship with Re (figures 10
and 11), the log Cp–log Re variation in figure 14 is almost
linear with a negative slope. Thus, for the range of Re
considered, the flow is in the viscous regime and pressure
varies with U. For flow past a symmetric bluff body, the
adverse pressure gradient generally commences at locations
close to its shoulders (see figure 7a and b for instance).
Thus, near the shoulders, the magnitude of velocity is low.
For AR ¼ 0:2, the flow around a shoulder becomes equiv-
alent to external flow past a corner. The cylinder surface is
a zero streamline, along which u ¼ 0 by virtue of the no-
1
2
3
4
5
6
0 10 20 30 40
Cp0
Re
AR = 0.2 0.5 0.8
0.5
1.0
1.5
2.0
0 10 20 30 401.0
1.2
1.4
1.6
1.8
2.0
0 10 20 30 40
a =
)c()b()a(
0 45 90
Figure 10. Steady unbounded flow past elliptic cylinders of AR ¼ 0:2, 0.5 and 0.8 for Re ¼ 5� 40: variation of the forward stagnation
pressure coefficient with Re for a ¼ (a) 0�, (b) 45� and (c) 90�.
0
1
2
3
0 10 20 30 40
-Cpb
Re
AR = 0.2 0.5 0.8
0.5
1.5
2.5
3.5
0 10 20 30 40 0.4
0.8
1.2
0 10 20 30 40
(a) (c)(b)
α = 0
0954
Figure 11. Steady unbounded flow past elliptic cylinders of AR ¼ 0:2, 0.5 and 0.8 for Re ¼ 5� 40: variation of the base suction
coefficient with Re for a ¼ (a) 0�, (b) 45� and (c) 90�.
Sådhanå (2020) 45:172 Page 11 of 18 172
-11
-4
0
4
8
0 90 180 270 360
Cp
θ (deg.)
Re = 1 3 6 7
-5
-4
-2
0
2
4
0 90 180 270 360
Re = 1 2
3.87
3.89
3.91
0 20 40 60
-4
-2
0
2
4
5
0 90 180 270 360
Re = 1
4.06
4.07
4.08
0 4 8
1.3
1.5
1.7
85 86 87 88
)c()b()a(
AR = 8.05.02.0
Figure 12. Steady unbounded flow past elliptic cylinders for a ¼ 90�: the surface pressure distribution at low Re for AR ¼ (a) 0.2, (b)
0.5 and (c) 0.8. For AR ¼ 0:2 at Re ¼ 7, it should be noted that the value of Cp at the local peak around h ¼ 86� (inset of figure a) is quitesmaller than the corresponding value of Cpfs summarized in table 1.
1
2
3
4
5
6
0 30 60 90
Cpf
s,C
pmax
α (deg.)
Re = 5, Cpfs Cpmax
Re = 40, Cpfs Cpmax
1.0
1.5
2.0
2.5
3.0
0 30 60 90
270
300
330
360
0 30 60 90
θ fs,
θ max
(de
g.)
α (deg.)
Re = 5, θfs θmax
Re = 40, θfs θmax
320
330
340
350
360
0 30 60 90
1.0
1.3
1.6
1.9
2.2
0 30 60 90
348
352
356
360
0 30 60 90
)c()b()a(
(d) (e) (f)
AR = 0.2
AR = 0.20.5
0.5 0.8
0.8
Figure 13. Steady unbounded flow past elliptic cylinders at Re ¼ 5 and 40: variation of the forward stagnation and maximum pressure
with a for AR ¼ (a) 0.2, (b) 0.5 and (c) 0.8. Also shown is the variation of h location of the forward stagnation and maximum pressure
points with a for AR ¼ (d) 0.2, (e) 0.5 and (f) 0.8.
172 Page 12 of 18 Sådhanå (2020) 45:172
slip condition. Thus, application of Bernoulli’s theorem
(though not applicable for real fluids, Bernoulli’s principle
provides useful estimates of velocity and pressure in real
flow) along this surface streamline cannot explain the dis-
crepancies between Cpfs and Cpmax. The streamlines next to
the w ¼ 0 streamline should therefore be considered to this
end. The streamlines upstream and downstream of the
corner, i.e. cylinder shoulder, undergo significant diversion
or bending. This excessive abrupt bending over an extre-
mely short extent (at the shoulder) is associated with a local
regime of low velocity around (ahead of and behind) the
shoulder. Thus, regimes of pressure of large magnitude, in
accordance with the principle of Bernoulli, exist in front of
and behind the shoulder. The pressure along the front half
of the cylinder is positive, and a negative or suction pres-
sure prevails along its rear half. For Re ¼ 1, the pressure at
the shoulders is negative and as Re increases, the h extent ofnegative pressure travels upstream. Large positive pressure
and large negative pressure, respectively, ahead and behind
the shoulder (see table 1) result in a discontinuity in Cp, i.e.
a positive and a negative pressure spike appear around
h ¼ 90�. The formation of the positive pressure spike is
responsible for the shift of the maximum pressure near the
shoulder rather than at the forward stagnation point. The
diversion of streamlines is most dominant at low Re regime
of attached flow, and both spikes are sharp. As Re increa-
ses, the average velocity of flow around the cylinder also
increases. Besides, a wake bubble first appears, then
enlarges and displaces its adjacent streamlines laterally
outwards. This effect reduces the bending of neighbour
streamlines. Because of the formation of wake, the average
velocity of flow behind the cylinder is smaller than that
ahead of the cylinder. The positive pressure spike is thus
eliminated or smoothed out, rendering the maximum pres-
sure to be attained at the forward stagnation point. This
leads to the identical values of Cpfs and Cpmax at higher Re.
For AR ¼ 0:5 and 0.8, the variation of velocity from near
the leading edge to the shoulders occurs in a gradual
manner. This appears to result in much lesser divergence in
h location of forward stagnation and maximum pressures
for these cylinders.
In summary, the angle by which the upstream
streamlines close to the cylinder shoulder get deflected
appears to govern whether Cpfs and Cpmax are identical or
different. For flow around the AR ¼ 0:2 cylinder at low
Re, the angle of diversion is very small (\\90�) and
Cpfs\Cpmax. As Re increases, the wake bubble expands
and the angle of diversion exceeds 90�. Consequently,Cpfs ¼ Cpmax. In case of a circular cylinder, the angle of
diversion is obtuse regardless of the flow being attached
or separated. In this case, Cpfs essentially represents
Cpmax for all Re.
7.3 Fluid loading
The dependence of drag coefficient on Reynolds number is
illustrated in figures 15a–c for cylinders of AR ¼ 0:2, 0.5,0.8 and 1. Three different values of a, i.e. 0�, 45� and 90�,are considered. Regardless of aspect ratio and angle of
attack, a monotonic decrease of Cd with Re is observed. For
a ¼ 0� and 45�, the Cd � Re curves for different cylinder
shapes are almost parallel and when Re is held constant, Cd
exhibits monotonic decrease with decreasing AR. However,
a contrasting feature is observed for a ¼ 90� (figure 15c)
where the effect of cylinder shape on Cd appears relatively
less significant. Figure 15d, e and f for, respectively, Re =
5, 20 and 40 indicates that Cd increases approximately
linearly with AR for small values of a. However, for large a,a slow decrease of Cd with increasing AR is apparent for
Re 20. The drag coefficient as a function of a is plotted in
figure 15g, h and i for Re ¼ 5, 20 and 40, respectively. For
each Re, drag of a cylinder increases monotonically with afrom its minimum at a ¼ 0� to the maximum at a ¼ 90�.This trend is consistent with the observations of Yano and
0.1
1
10
1 10 100
Cp0
, Cpb
Re
Cp0Cpb
0.1
1
10
1 10 100
Cp0
, Cpb
Re
Cp0Cpb
0.2 0.8
(b)(a)
AR =
Figure 14. Variation of Cp0 and Cpb with Re in log–log plane for steady unbounded flow past elliptic cylinders of AR ¼ (a) 0.2 and (b)
0.8 both at 0� incidence.
Sådhanå (2020) 45:172 Page 13 of 18 172
Kieda [27] for Re� 1 and Dennis and Young [4] for a thin
cylinder at Re ¼ 5, 20 and 40. At low Re, such as 5, the
drag of a circular cylinder overshadows those of its elliptic
counterparts of any orientation. Also, over a considerable
fraction of the range of a, the drag on an elliptic cylinder
for Re 20 (figure 15h, i) is less than the one for a circular
cylinder. The angle of attack at which elliptic cylinders of
various AR generate approximately the same drag has an
inverse relationship with Re. For instance, at Re ¼ 10,
cylinders of various thicknesses give the same drag when
a ¼ 90� (figure 15c). For Re ¼ 20 and 40, these angles
approximately equal 75� (figure 15h) and 60� (figure 15i),
respectively.
For a streamlined body, Cdv is dominant over Cdp. The
value ofCdp
Cdvratio for such bodies, therefore, falls below
unity. In contrast, for a bluff body, the value ofCdp
Cdvratio
exceeds unity. For Re ¼ 5, 20 and 40, figure 16a, b and c,
respectively, plots the relationship between theCdp
Cdvratio and
a. For each AR and Re, theCdp
Cdvratio increases monotonically
as a continues to rise. For Re ¼ 5, the value ofCdp
Cdvratio for
each AR is virtually constant at unity for a � 34�. For a
smaller than this angle,Cdp
Cdv\1, signifying that the stream-
lined nature dominates; for higher values of a, the cylinders
tend to become more bluff. When Re ¼ 20, theCdp
Cdvratio
turns invariant to AR for a � 30�. For elliptic cylinders withAR� 0:5, the streamlined nature becomes predominant for
values of a smaller than 26�, approximately. For Re ¼ 40,
the angle at which the ratio of pressure to viscous drag
0.5
1.5
2.5
3.5
4.5
0 10 20 30 40
Cd
Re
AR = 0.2 0.5 0.8
1
0.5
1.5
2.5
3.5
4.5
0 10 20 30 40 0.5
1.5
2.5
3.5
4.5
0 10 20 30 40
2.4
3.0
3.6
4.2
0.2 0.4 0.6 0.8 1
Cd
AR
α = 0ο
15ο
45ο
75ο
90ο
1.0
1.4
1.8
2.2
0.2 0.4 0.6 0.8 10.6
0.9
1.2
1.5
1.8
0.2 0.4 0.6 0.8 1
2.6
3.0
3.4
3.8
4.2
0 30 60 90
Cd
α (deg.)
AR = 0.2 0.5 0.8
1
1.0
1.4
1.8
2.2
0 30 60 900.7
1.0
1.3
1.5
1.7
0 30 60 90
(a)
)h()g(
)f()e()d(
)c()b(
α = 0
Re = 50402
0954
20 40Re = 5
(i)
Figure 15. Steady unbounded flow past elliptic cylinders of aspect ratios 0.2, 0.5, 0.8 and 1: variation of total drag with Re for a = (a)
0�, (b) 45� and (c) 90�; with AR for Re ¼ (d) 5, (e) 20 and (f) 40; with a for Re ¼ (g) 5, (h) 20 and (i) 40.
172 Page 14 of 18 Sådhanå (2020) 45:172
components displays insensitivity to Re lowers down to
about 30�. It may be noted that the value ofCdp
Cdvratio at this
angle exceeds unity. With AR decreasing, the ratio at a
given angle of attack decreases for a� 20� and increases
for a 45�. Roughly the neighbourhood of 20� represents acritical zone of angle of attack, below which a cylinder (for
AR� 0:5) tends to behave like a streamlined body and
above, like a bluff body. Therefore, the regime of a over
which an elliptic cylinder dominantly exhibits the features
of a bluff body rather than a streamlined body increases
with Re. However, for Re ¼ 5; 20 and 40 each, the con-
figurations of a ¼ 90� and 0�, respectively, correspond to
the maximum and minimum bluffness of an elliptic cylin-
der. At the terminal values of a, i.e. a ¼ 90� and 0�,respectively, the maximum bluff and maximum streamlined
behaviours are associated with the AR ¼ 0:2 elliptic
cylinder. An opposite trend is displayed by the thick
cylinder of AR ¼ 0:8. For a ¼ 90� configuration, figure 16d
shows the wake of the AR ¼ 0:2 cylinder at a very low Re
value of 1.07 further underscoring the fact that the maxi-
mum bluff behaviour is associated with this cylinder.
A cylinder placed symmetrically, relative to the free-
stream, experiences no net lift or moment. This feature is an
outcome of symmetric surface pressure (figure 5a, c) and
anti-symmetric surface vorticity (figure 3a, c) distributions
about the base point. Figure 17a, b and c illustrates the
dependence of Cl on Re for a = 15�, 45� and 75�, respec-tively. A monotonically decreasing Cl � Re relationship
that depends strongly on the cylinder shape is observed. For
a thick cylinder (AR = 0.8), the decay of Cl with Re is
insignificant. In contrast, Cd shows a strong decrease.
Figure 17d–f demonstrates, for moderate to higher values
of a, approximate linear decrease of Cl with increasing AR.
The Cl � a variation for Re ¼ 5, 20 and 40 is shown,
respectively, in figure 17g, h and i for AR = 0.2, 0.5 and 0.8.
As a increases from 0� to 90�, Cl exhibits non-monotonic
variation with a, i.e. initially increases from and then
decreases to zero. Similar variation of Cl with a was earlier
(a) (b)
(c) (d)
Figure 16. Steady unbounded flow past elliptic cylinders of AR ¼ 0.2, 0.5 and 0.8: variation of the Cdp=Cdv ratio with a for Re ¼ (a) 5,
(b) 20 and (c) 40. The approximate boundary of a, roughly demarcating the regimes of streamlined and bluff behaviours, is also shown. It
may be noted that this boundary shifts towards lower a as Re is increased progressively. In terms of streamlines, figure d shows the closed
wake of the AR ¼ 0:2 cylinder for a ¼ 90� at Re ¼ 1:07.
Sådhanå (2020) 45:172 Page 15 of 18 172
noted by Yano and Kieda [27] and Dennis and Young [4].
Irrespective of Re, the maxima in Cl for AR 0:5 occur at
a ¼ 45� among the angles of attack studied. For the thin
cylinder, however, the maximum is attained at a ¼ 45� forRe ¼ 5 and at a approximately 30� for Re ¼ 20; 40. It isalso observed that the Cl � a curves for AR ¼ 0:8 are
nearly symmetric about a ¼ 45�. With decreasing AR, lack
of symmetry about a ¼ 45� becomes prominent; Cl pre-
dicted at a\45� overshadows the corresponding value at
90� � a, i.e. the maximum lift shifts to \45�.For Re ¼ 40, table 2 summarizes the values of aerody-
namic coefficients along with their pressure and viscous
components. For symmetric elliptic cylinders, the mean lift
is zero. The viscous components of lift cancel out for such
orientations (opposite signed between the upper and lower
surfaces). For other orientations, the surface pressure
between the stagnation points displays asymmetry; hence,
lift is non-zero. For such cases, negative sign is associated
with the viscous component of lift force, Clv. The negative
or downward acting Clv was earlier predicted by Lugt and
Haussling [14] and Dennis and Young [4]. It is found that
Clp is an order of magnitude higher than Clv.
8. Conclusion
A stabilized finite-element method has been employed to
investigate the aerodynamic loading for steady, laminar
flow around stationary elliptic cylinders at various angles of
attack. The aspect ratios studied are 0.2, 0.5 and 0.8. The
aerodynamic behaviour is studied for Re� 40 and
0� � a� 90�. With major axis of the cylinder normal to the
flow, the forward stagnation pressure for AR ¼ 0:2, Re ¼1� 6 and AR ¼ 0:5, Re ¼ 1 is always smaller than the
maximum pressure occurring at/near the cylinder shoulders.
The discrepancy between Cpfs and Cpmax exists for low Re
flow around cylinders having abrupt changes in body con-
tour. With increasing Re and AR, the forward stagnation
pressure represents the maximum pressure for both the
symmetric (here, a ¼ 90�) and asymmetric configurations.
The angle of diversion of the upstream streamlines close to
the cylinder shoulder appears to govern the occurrence of
this phenomenon. For Re on the order of unity, diversion by
small angles renders Cpfs to fall short of Cpmax; these
quantities become identical when the angle of diversion is
obtuse, as in a circular cylinder. The closeness of forward
stagnation and maximum pressure, however, does not
imply that they occur at the same location on the surface of
an asymmetric cylinder. The forward stagnation and max-
imum surface pressure of a cylinder continue to decrease
with increasing a. For symmetric cylinders, the magnitude
of minimum surface pressure decreases with increasing Re.
While the location of minimum pressure, or alternatively
onset of adverse pressure gradient, travels upstream for
AR ¼ 0:5 and 0.8, the location is fixed at the base point (for
a ¼ 0�) or shoulders (for a ¼ 90�) for AR ¼ 0:2. When Re
and a are both fixed, the base suction coefficient generally
decreases with increasing AR. With increasing Re, the base
Table 2. Steady unbounded flow past elliptic cylinders of 0:2�AR� 0:8: summary of the aerodynamic coefficients and their com-
ponents at Re ¼ 40 for various angles of attack between 0� and 90�. It may be noted that the moment disappears when the lift is zero.
AR a Cdp Cdv Cd Clp Clv Cl Cmp Cmv Cm
0.2 0� 0.1648 0.6105 0.7753 0 0 0 0 0 0
15� 0.3106 0.5381 0.8487 0.6393 –0.0525 0.5868 0.1664 0.0132 0.1796
30� 0.6329 0.4043 1.0372 0.9055 –0.0748 0.8306 0.2372 0.0151 0.2523
45� 0.9662 0.2969 1.2631 0.8567 –0.0810 0.7757 0.2205 0.0109 0.2314
60� 1.2394 0.2204 1.4598 0.6480 –0.0702 0.5778 0.1613 0.0066 0.1679
75� 1.4299 0.1784 1.6083 0.3512 –0.0516 0.2996 0.0826 0.0049 0.0875
90� 1.4792 0.1559 1.6351 0 0 0 0 0 0
0.5 0� 0.4488 0.5840 1.0328 0 0 0 0 0 0
15� 0.5154 0.5572 1.0726 0.3211 –0.0179 0.3032 0.1121 0.0099 0.1220
30� 0.6842 0.4945 1.1787 0.5038 –0.0297 0.4741 0.1769 0.0139 0.1908
45� 0.8901 0.4264 1.3165 0.5196 –0.0344 0.4852 0.1820 0.0123 0.1943
60� 1.0749 0.3708 1.4457 0.4111 –0.0308 0.3803 0.1423 0.0084 0.1507
75� 1.2001 0.3352 1.5353 0.2247 –0.0183 0.2064 0.0768 0.0041 0.0809
90� 1.2443 0.3228 1.5671 0 0 0 0 0 0
0.8 0� 0.7669 0.5473 1.3142 0 0 0 0 0 0
15� 0.7891 0.5398 1.3289 0.1081 –0.0052 0.1029 0.0476 0.0035 0.0511
30� 0.8487 0.5202 1.3689 0.1814 –0.0091 0.1723 0.0798 0.0057 0.0855
45� 0.9274 0.4951 1.4225 0.2014 –0.0107 0.1907 0.0885 0.0060 0.0945
60� 1.0032 0.4719 1.4751 0.1682 –0.0095 0.1587 0.0737 0.0047 0.0784
75� 1.0570 0.4558 1.5128 0.0947 –0.0056 0.0891 0.0414 0.0026 0.0440
90� 1.0764 0.4501 1.5265 0 0 0 0 0 0
172 Page 16 of 18 Sådhanå (2020) 45:172
suction continues to decay. As a increases from 0� to 90�,irrespective of AR, the drag coefficient increases mono-
tonically while lift and moment increase from and then
decrease to zero. As AR increases to unity, the lift and
moment decrease monotonically to zero. In contrast, the
drag increases or decreases slowly depending on Re and a.The drag of a circular cylinder at low Re is more than that
generated by an elliptic cylinder. As Re increases, this trend
prevails over a wide range of a. The regime of a over whichan elliptic cylinder displays the features of a bluff body,
more dominantly than those of a streamlined body,
increases with Re. For Re ¼ 40, elliptic cylinders of
AR� 0:5 behave more like a streamlined body for a\20�
and a bluff body for a 20�.
Nomenclaturea Major axis of ellipse
b Minor axis of ellipse
AR Aspect ratio of ellipse
Cd Drag coefficient
Cdp Pressure drag coefficient
Cdv Viscous drag coefficient
Cl Lift coefficient
Clp Pressure lift coefficient
Clv Viscous lift coefficient
Cm Moment coefficient
Cmp Pressure component of moment coefficient
Cmv Viscous component of moment coefficient
Cp Pressure coefficient
0.0
0.4
0.8
1.2
0 10 20 30 40
Cl
Re
AR = 0.2 0.5 0.8
0.0
0.4
0.8
1.2
0 10 20 30 400.0
0.4
0.8
1.2
0 10 20 30 40
0.0
0.4
0.8
1.2
0 0.2 0.4 0.6 0.8 1
Cl
AR
α = 15ο
30ο
45ο
60ο
75ο
0.0
0.4
0.8
1.2
0 0.2 0.4 0.6 0.8 10.0
0.4
0.8
1.2
0 0.2 0.4 0.6 0.8 1
0.0
0.4
0.8
1.2
0 30 60 90
Cl
α (deg.)
AR = 0.2 0.5 0.8
0.0
0.4
0.8
1.2
0 30 60 900.0
0.4
0.8
1.2
0 30 60 90
(a) (c)(b)
5754
(d) (f)(e)
Re = 0402 Re = Re = 5
Re = 20
(g)
= α = αα = 15
5Re = Re = 40
(i)(h)
Figure 17. Steady unbounded flow past elliptic cylinders of AR ¼ 0:2, 0.5, 0.8 and 1: variation of total lift with Re for a ¼ (a) 15�, (b)45� and (c) 75�; with AR for Re ¼ (d) 5, (e) 20 and (f) 40; with a for Re ¼ (g) 5, (h) 20 and (i) 40.
Sådhanå (2020) 45:172 Page 17 of 18 172
Cpb Base pressure coefficient
Cpfs Forward stagnation pressure coefficient
Cpmax Maximum value of surface pressure coefficient
Cpmin Minimum value of surface pressure coefficient
h Surface traction vector
m Connectivity of flow section
n Unit normal to the surface
N Number of four-way nodes
N0
Number of three-way nodes
p Pressure
q Weight function for pressure
Re Reynolds number
S Number of four-way saddles
S0
Number of three-way saddles
u Velocity vector
U Free-stream speed
x Spatial coordinate vector
w Weight function vector for velocity components
a Angle of attack
d Stabilization parameter
� Strain rate tensor
q Density of fluid
w Streamfunction
x Vorticity
h Circumferential angle
r Stress tensor
X Domain
C Domain boundary
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