Drag coefficient calculator and table of common shapes
Calculate the drag and lift coefficients from a force you have measured or simulated, or the force from a coefficient, and look up CD in the table of common shapes, with the Reynolds numbers where each value holds.
Drag and lift coefficient calculator
It opens on a 100 mm sphere in air at 10 m/s, works as a drag force calculator too, and its second tab is a terminal velocity calculator.
1Fluid
Viscosity and speed of sound
2Flow and body
2D gives forces per metre of span, as in a 2D simulation.
Table value CD 0.47 (White, Table 7.3); valid from Re 10⁴ to about 2 × 10⁵; frontal area, Re from the diameter.
Re uses L = 100.0 mm from the diameter until you type a length.
3Forces and coefficients
Check a Simcenter STAR-CCM+ coefficient
Type the coefficient from your Force Coefficient report. Add its reference values and the check names the one that differs.
A new unit keeps the digits you typed; the SI/US switch and the examples convert. A decimal comma works too.
2Sphere
A new unit keeps the digits you typed; the SI/US switch and the examples convert. A decimal comma works too.
Result
Example: sphere in air
Drag coefficient CD0.470
- Dynamic pressure q
- 61.3 Pa
- Force per unit coefficient, qA
- 0.481 N
- Reynolds number Re
- 6.85 × 10⁴ (Newton plateau)
- Mach number M
- 0.029
Reference: A = 0.007854 m² (frontal, circle 100 mm), ρ = 1.225 kg/m³, U = 10.0 m/s.
- Passed: M 0.029: incompressible
- Passed: Re 6.85 × 10⁴ is inside the table range (10⁴ to 2 × 10⁵)
- Note: Smooth sphere at this Re: 0.49
Forces on the body
Drag coefficient against Reynolds number
Show the data
The plot shows the smooth-sphere curve for a sphere, and for the circular cylinder for comparison, or for any result with both CD and Re.
Show the calculation with your numbers
Reference values for the Force Coefficient report
The report divides the force by ½ ρ v² a with the reference values typed into it; the solver does not check them against the inflow. A Frontal Area report gives the projected area.
Coefficients use q = ½ρU². Table values and correlations hold within their stated ranges; above M = 0.3, where p₀ − p is over 2 % above q, the tool warns. Free of charge and without warranty.
Drag coefficient formula and the drag equation
The drag coefficient CD is the drag force divided by the dynamic pressure of the flow and a reference area. It has no units, which means that a value you measure on a model in a wind tunnel holds for the full-size body, as long as the body is geometrically similar, the reference area is defined the same way and the Reynolds and Mach numbers match [2].
How to calculate the drag coefficient
CD = FD / (qA) = 2FD / (ρU2A) (1)
q = ½ρU2 (2)
Here FD is the drag force (N), the part of the force on the body along the free stream, ρ the fluid density (kg/m³), U the flow speed relative to the body (m/s, often written v or V) and A the reference area (m²). The dynamic pressure q (Pa) is the pressure rise you get when a low-speed flow is brought to rest without losses.
Take the example the calculator opens with, a sphere of 100 mm diameter held in air at 15 °C and sea-level pressure, where ρ = 1.225 kg/m³ [15], with a flow speed of 10 m/s. Say that a balance, or a Force report in Simcenter STAR-CCM+, gives you a drag force of 0.226 N. The dynamic pressure is then q = ½ × 1.225 × 10² = 61.25 Pa, the frontal area is π × 0.1²/4 = 7.854 × 10⁻³ m², and equation (1) gives CD = 0.226 / (61.25 × 7.854 × 10⁻³) = 0.470. The Reynolds number is 6.85 × 10⁴, inside the range where a smooth sphere has a CD of about 0.47 (see the sphere section below).
Drag force from the drag equation
FD = ½ρU2CDA (3)
Turned around, the definition gives the drag force when you know CD. This is the drag equation [3], also called the drag force formula. The force grows with the square of the speed, and twice the speed gives four times the drag, while the power you need to overcome it, P = FDU, grows with the cube.
As an example, take a car shape with CD = 0.30 and a frontal area of 2.2 m² (example values, not a particular car) at 100 km/h in air at 15 °C. The dynamic pressure is 473 Pa, the drag force 312 N and the power to overcome it about 8.7 kW.
Dynamic pressure and the low-speed limit
At 30 m/s in the same air, q = ½ × 1.225 × 30² = 551 Pa. Note that q = ½ρU2 is a definition and holds at any speed, but the tables and the sphere curve on this page are low-speed data. Above a Mach number of about 0.3 (about 100 m/s in air at 15 °C) compressibility starts to show, and the calculator warns from there. At M = 0.3 the pressure rise to the stagnation point, p0 − p, is already 2.3 % larger than q [9].
Drag has two parts, pressure drag (form drag, from the pressure acting on the body) and friction drag (from the shear stress in the boundary layer). On a bluff body, such as the sphere you calculated above, the flow separates and leaves a wide wake at low pressure, and pressure drag dominates. On a streamlined body the flow stays attached and friction is a large part of the total. In Simcenter STAR-CCM+ you can split the two with the Force Option of the force reports, which takes pressure, shear or both [14].
Which reference area?
When you quote a CD, say which area it is based on, since the same force gives a different coefficient for each choice [3]. Bluff bodies and vehicles use the frontal area (the area projected on a plane normal to the flow). Wings use the planform area (the wing seen from above), and the Simcenter STAR-CCM+ documentation gives the planform area for an airfoil and the projected frontal area for vehicles and other objects [14]. Some data use the wetted area, the whole surface of the body [3]. A 2D section uses a length per unit span, the chord or the frontal height, and its coefficients are written in lower case, cd and cl.
One wing, one drag force, four reference areas. Shaded areas are drawn to the same scale (span b = 4c).
Frontal area
seen from upstream; bluff bodies, vehicles
A = t b = 0.12 c b
CD = 0.067
Planform area
seen from above; wings
A = c b
CD = 0.0081
Wetted area
the whole skin; ships, friction drag
A ≈ 2.04 c b
CD = 0.0040
Unit span (2D)
per metre of span; 2D CFD, airfoil sections
A = c × 1 m
cd = 0.0081
Same force in every card: CD × A = 0.0081 c b; the frontal value is about 8 times the planform value for the same wing. Example: a NACA 0012 wing at α = 0° with the measured section drag cd = 0.0081 at Re = 6 × 10⁶ [10]; t/c = 0.12 and the section perimeter, 2.04 c, from the NACA 4-digit thickness formula [18]; wing tips left out.
Two coefficients can only be compared on the same area, and you convert between them with the ratio of the areas [3]. The sphere in the example has a wetted area of πd², four times its frontal area, and the same 0.226 N then gives CD = 0.117 on the wetted area, a quarter of the frontal value.
Drag coefficient tables
The tables give CD on the frontal area, with the flow from left to right. The values are from White [1] unless a row says otherwise, and White gives them for Re ≥ 10⁴. Shapes with sharp edges separate at the edges, which keeps their value nearly constant above that. Rounded shapes (the circular cylinder, the sphere, and the rounded sides of the half cylinder and the cup) keep it only below the drag crisis, and the tables use about Re = 2 × 10⁵ as a safe upper limit for them (see the sphere and cylinder sections). The 2D values are per unit span, on the frontal height. The Use links load a row into the calculator with its area convention.
2D shapes
| Shape (flow from left to right) | cd | Holds for | Source | Calculator |
|---|---|---|---|---|
| Circular cylinder | 1.2 | Re 10⁴ to about 2 × 10⁵ (below the drag crisis) | [1] Table 7.2 | Use Circular cylinder in the calculator ↑ |
| Half cylinder, rounded side facing the flow | 1.2 | Re 10⁴ to about 2 × 10⁵ (below the drag crisis) | [1] Table 7.2 | Use Half cylinder, rounded side facing the flow in the calculator ↑ |
| Half cylinder, flat side facing the flow | 1.7 | Re above 10⁴ (sharp edges) | [1] Table 7.2 | Use Half cylinder, flat side facing the flow in the calculator ↑ |
| Square cylinder, face to the flow | 2.1 | Re above 10⁴ (sharp edges) | [1] Table 7.2 | Use Square cylinder, face to the flow in the calculator ↑ |
| Square cylinder at 45°, corner to the flow | 1.6 | Re above 10⁴ (sharp edges) | [1] Table 7.2 | Use Square cylinder at 45°, corner to the flow in the calculator ↑ |
| Flat plate across the flow | 2.0 | Re above 10⁴ (sharp edges) | [1] Table 7.2 | Use Flat plate across the flow in the calculator ↑ |
| Equilateral triangle, flat side facing the flow | 2.0 | Re above 10⁴ (sharp edges) | [1] Table 7.2 | Use Equilateral triangle, flat side facing the flow in the calculator ↑ |
| Equilateral triangle, tip facing the flow | 1.6 | Re above 10⁴ (sharp edges) | [1] Table 7.2 | Use Equilateral triangle, tip facing the flow in the calculator ↑ |
3D shapes
| Shape (flow from left to right) | CD | Holds for | Source | Calculator |
|---|---|---|---|---|
| Sphere (smooth) | 0.47 | Re 10⁴ to about 2 × 10⁵ (below the drag crisis) | [1] Table 7.3 | Use Sphere (smooth) in the calculator ↑ |
| Disc, face to the flow | 1.17 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Disc, face to the flow in the calculator ↑ |
| Cube, face to the flow | 1.07 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cube, face to the flow in the calculator ↑ |
| Cube, edge to the flow | 0.81 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cube, edge to the flow in the calculator ↑ |
| Hollow hemisphere (cup), open side facing the flow | 1.4 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Hollow hemisphere (cup), open side facing the flow in the calculator ↑ |
| Hollow hemisphere (cup), rounded side facing the flow | 0.4 | Re 10⁴ to about 2 × 10⁵ (below the drag crisis) | [1] Table 7.3 | Use Hollow hemisphere (cup), rounded side facing the flow in the calculator ↑ |
| Cone, tip facing the flow, by angle between the sides | ||||
| 10° between the sides | 0.3 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cone, tip facing the flow, by angle between the sides, 10° between the sides, in the calculator ↑ |
| 20° between the sides | 0.4 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cone, tip facing the flow, by angle between the sides, 20° between the sides, in the calculator ↑ |
| 30° between the sides | 0.55 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cone, tip facing the flow, by angle between the sides, 30° between the sides, in the calculator ↑ |
| 40° between the sides | 0.65 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cone, tip facing the flow, by angle between the sides, 40° between the sides, in the calculator ↑ |
| 60° between the sides | 0.8 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cone, tip facing the flow, by angle between the sides, 60° between the sides, in the calculator ↑ |
| 75° between the sides | 1.05 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cone, tip facing the flow, by angle between the sides, 75° between the sides, in the calculator ↑ |
| 90° between the sides | 1.15 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cone, tip facing the flow, by angle between the sides, 90° between the sides, in the calculator ↑ |
| Cylinder along the flow, flat ends, by length/diameter | ||||
| L/d = 0.5 | 1.15 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cylinder along the flow, flat ends, by length/diameter, L/d = 0.5, in the calculator ↑ |
| L/d = 1 | 0.9 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cylinder along the flow, flat ends, by length/diameter, L/d = 1, in the calculator ↑ |
| L/d = 2 | 0.85 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cylinder along the flow, flat ends, by length/diameter, L/d = 2, in the calculator ↑ |
| L/d = 4 | 0.87 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cylinder along the flow, flat ends, by length/diameter, L/d = 4, in the calculator ↑ |
| L/d = 8 | 0.99 | Re above 10⁴ (sharp edges) | [1] Table 7.3 | Use Cylinder along the flow, flat ends, by length/diameter, L/d = 8, in the calculator ↑ |
| Streamlined body (airfoil-shaped model, frontal area) | 0.045 | Low-speed wind tunnel, Re not given | [2] | Use Streamlined body (airfoil-shaped model, frontal area) in the calculator ↑ |
The streamlined body is NASA Glenn’s value from low-speed wind tunnel tests on the projected frontal area [2]. A wing is normally given on its planform area, which is many times its frontal area, and you cannot use the 0.045 of the streamlined body for a wing. For the sphere, the standard drag curve stays between about 0.39 and 0.50 from Re ≈ 10³ to 2 × 10⁵ [5], a little wider than the table’s range.
Drag coefficient of a sphere
A smooth sphere has a drag coefficient of about 0.47 on its frontal area for Reynolds numbers from 10⁴ to about 2 × 10⁵. White gives 0.47 for Re ≥ 10⁴ [1], and the standard drag curve stays between about 0.39 and 0.50 from Re ≈ 10³ up to 2 × 10⁵ [5]. At very low Reynolds numbers CD = 24/Re (Stokes’ law) [11], and near Re = 3.7 × 10⁵ the drag crisis takes it below 0.1 [5], [6].
The Reynolds number, which tells you where on the curve you are, is Re = ρUd/μ, with d the diameter and μ the dynamic viscosity of the fluid.
Drag coefficient vs Reynolds number
The figure shows CD against Re for a smooth sphere and a long circular cylinder. At low Re the flow creeps around the body, viscous forces dominate and the sphere’s CD falls as 24/Re. From about Re = 10³ the boundary layer separates near the widest point while it is still laminar, the wake is wide and CD stays nearly flat (the Newton plateau, or subcritical range). At the drag crisis the boundary layer turns turbulent before it separates, and separation moves to the rear, where the narrower wake gives a lower CD. In the large-eddy simulations of Li et al., the separation point on a smooth sphere moved from 79° to 102° from the front stagnation point between the subcritical and the supercritical range [8].
- 1 Stokes
- 2 Intermediate
- 3 Newton plateau
- 4 Drag crisis
- 5 Supercritical
The sphere curve in the figure and in the calculator is the standard drag curve of Clift, Grace and Weber [5]. We use it up to Re = 10⁶ and draw it only there.
Drag crisis
The drag crisis is the sharp drop in CD when the boundary layer turns turbulent before it separates, and it does not sit at a fixed Reynolds number. Achenbach’s 3.7 × 10⁵ is for a smooth sphere in a wind tunnel [6], and roughness and the test conditions move it [7], [8]. Near the crisis a measured CD can differ from the curve by a factor of two or more, since the curve falls from about 0.47 near Re = 3.4 × 10⁵ to 0.09 at 4 × 10⁵ [5].
Drag coefficient of a cylinder
A long circular cylinder across the flow is the 2D case. White gives cd = 1.2 on the diameter, per unit span, while the boundary layer is laminar, and 0.3 when it is turbulent [1]. As for the sphere, the drop comes at about 3 × 10⁵ for a smooth cylinder [1], [17]. A shorter cylinder has less drag, because flow escapes around the ends, and White gives 0.64 for a length of one diameter, 0.82 for ten and 0.98 for forty [1].
Behind the cylinder, vortices shed from alternate sides (the von Kármán vortex street in our simulation further down), and if you plot the two reports you see the lift swing around zero while the drag oscillates around its mean.
Surface roughness
Roughness trips the boundary layer into turbulence earlier, and a rough sphere reaches the drag crisis at a lower Reynolds number than a smooth one. Above the crisis its drag is higher than that of the smooth sphere [7].
Example: why golf balls have dimples
Golf balls have dimples, which act as roughness [8]. Take a ball of 42.7 mm, the size Li et al. simulated [8], at 70 m/s in air at 15 °C (our example speed). Its Reynolds number is 2.0 × 10⁵, just below the crisis of a smooth sphere, where CD is still about 0.5 [5].
The dimples trip the boundary layer, and in the simulations of Li et al. the golf ball reached turbulent separation at a lower Reynolds number than the smooth sphere. Its separation point moved from 80° to about 105°, and at Re = 1.1 × 10⁵ its drag coefficient was about 0.22, against 0.46–0.47 for the smooth sphere in the subcritical range [8]. At our example speed the dimpled ball is past its crisis.
The lift on a golf ball comes from its backspin (the Magnus effect), not from the dimples on their own.
Lift coefficient and the lift equation
Lift is the part of the force on a body perpendicular to the free stream, and drag is the part along it [14]. Note that this is the free-stream direction, not the body’s own axes. On a wing at an angle of attack α, lift is not normal to the chord, and drag is not along it. The force diagram in the calculator draws drag along the flow and lift across it, with the resultant between them.
The lift equation
FL = ½ρU2CLA (4)
CL = 2FL / (ρU2A) (5)
The lift equation (the lift formula) has the same form as the drag equation, and NASA Glenn writes it with the wing area [4]. For a wing, A is the planform area S, and both coefficients of an aircraft are normally given on S. The lift-to-drag ratio L/D = CL/CD is then the ratio of the two forces.
The wing example in the calculator works backwards from the lift. Take a light aircraft weighing 12 kN with 16 m² of wing and a maximum lift coefficient of 1.5 (example values). In level flight the lift equals the weight, and equation (4) solved for the speed gives the slowest speed it can fly at sea level, U = √(2FL / (ρSCL,max)) = 28.6 m/s, or 55.5 knots, below which the wing cannot carry the weight.
Load the wing example in the calculator ↑
Lift coefficient against angle of attack
For a thin airfoil at small angles, thin airfoil theory gives cl = 2π(α − αL0), with α in radians and αL0 the zero-lift angle, which is zero for a symmetric section [9]. In degrees the slope is 0.110 per degree, which gives cl ≈ 0.55 for a symmetric section at 5°. Measured sections follow this slope closely until the flow starts to separate from the upper surface. Then the curve bends over and reaches cl,max, and beyond that the lift drops and the section stalls. For the NACA 0012 in the figure, measured at Re = 6 × 10⁶, the maximum is about 1.61 at 17° [10].
Note that the 2π slope is for a 2D section (cl per unit span) in attached flow. That is why the calculator does not give CL from α. It takes the coefficient from you, from a test, a simulation or a figure like this one.
Finite wings and induced drag
A wing of finite span loses lift towards the tips, where air flows around from the high-pressure side to the low-pressure side and leaves trailing vortices. Prandtl’s lifting-line theory gives the lift slope of the whole wing and the induced drag that comes with the lift [9].
CL,α = a0 / (1 + a0 / (πeAR)) (6)
CD,i = CL2 / (πeAR) (7)
Here a0 is the section slope (2π per radian), AR = b²/S the aspect ratio (span squared over wing area) and e the span efficiency factor, 1 for an elliptical lift distribution and lower for other wings. Strictly, the factors in (6) and (7) are two different ones that are both 1 for elliptical loading. The theory holds for unswept wings with an aspect ratio above about 4 in attached flow [9].
With AR = 8 and e = 0.9 the slope drops from 0.110 to 0.086 per degree, and at CL = 0.5 the induced drag coefficient is 0.011. Induced drag grows with the square of CL, which makes it largest at low speed, when the wing needs a high CL to carry the weight.
Terminal velocity of a falling sphere
A sphere falling through a fluid speeds up until drag and buoyancy balance its weight. At that terminal velocity vt, ½ρvt²CD(πd²/4) = (ρp − ρ)g(πd³/6), with ρp the density of the sphere and ρ that of the fluid. If ρp is lower than ρ, the sphere rises instead. This is for a solid sphere, since bubbles and drops deform and circulate inside, and they rise or fall differently.
Terminal velocity formula
Solved for the speed, the force balance gives the terminal velocity formula.
vt = √(4gd(ρp − ρ) / (3ρCD)) (8)
Since CD depends on the Reynolds number, which in turn depends on vt, formula (8) has to be solved together with the sphere curve, and the falling-sphere tab of the calculator does this for you.
Stokes’ law
At very low Reynolds numbers CD = 24/Re, and formula (8) becomes Stokes’ law [11].
vt = (ρp − ρ)gd² / (18μ) (9)
If you take a 2 mm steel ball (7 850 kg/m³) in water at 20 °C [16], Stokes’ law gives 14.9 m/s, at a Reynolds number of about 3 × 10⁴, which is far outside the range where it holds, while the full sphere curve in the calculator gives 0.63 m/s at Re = 1.3 × 10³, where CD ≈ 0.45. Stokes’ law is within 2 % of the standard curve only below Re ≈ 0.1 [5].
Several solutions near the drag crisis
Near the drag crisis the drag force falls as the speed rises, between Reynolds numbers of about 3.4 × 10⁵ and 4.0 × 10⁵, and the force balance can then have three solutions. For a 43.7 mm steel ball falling in air at 15 °C, the curve gives 86.6, 125.7 and 203.5 m/s, and the middle one is unstable. A ball dropped from rest reaches the lowest one and stays there. The calculator reports that one and lists the others. The two higher speeds are above Mach 0.3, where the low-speed curve is only an estimate.
Simulating drag and lift in Simcenter STAR-CCM+
The picture below is from one of our own Simcenter STAR-CCM+ simulations, the von Kármán vortex street behind a cylinder, with the flow from left to right. The mesh pictures further down are of a cylinder too.
Above a Reynolds number of about 190 the wake behind a cylinder becomes three-dimensional [13]. A 2D simulation cannot follow that, and in our experience it then gives too much drag. Use it to learn the set-up, and a 3D model for numbers you will rely on.
Computational domain
Size the domain in diameters of the body. Behr et al. found that the side boundaries matter more than is commonly assumed. They computed the flow past a cylinder at Re = 100, with symmetry conditions on the sides, and recommend keeping the side boundaries at least 8 diameters from the cylinder, since the mean drag coefficient came out up to 6 % too high with the boundaries closer [12]. Upstream we usually leave about 20 diameters, which keeps the inlet from forcing the flow in front of the body, and downstream about 60, which keeps the outlet from disturbing the wake.
Mesh
Refine where the gradients are, near the cylinder and in the wake, and keep the cells coarser elsewhere. To estimate how much CD and CL still depend on the mesh, run three systematically refined meshes and put the results into the grid convergence index calculator.
(a) Edge of the wake refinement.
(b) Local refinement around the cylinder.
(c) Polyhedral cells meeting the prism layer.
(d) Prism layers at the wall.
A wake refinement grows from a part surface or a part curve, and you then set the cell size, the length of the refined region and, as an expert property, its spread angle. Panel a shows its upper edge, where the small cells inside the refinement grow into the coarser cells outside it.
Near the wall we aim for a first cell with y+ below 1, enough prism layers to cover the boundary layer, a stretching of 1.1–1.2 and a last prism layer about as thick as the first core cell (panels c and d). The y+ calculator sizes all of these. A y+ below 1 matters most where no edge fixes the separation point, as on the cylinder and the sphere. On a body with sharp edges the flow separates at the edges, and the drag is less sensitive to the wall treatment.
Turbulence model
We usually start external aerodynamics with the k–ω SST model (the turbulence model selector lists the alternatives for your case). A fully turbulent RANS model puts a turbulent boundary layer on the front of a smooth cylinder or sphere even in the subcritical range, where the real boundary layer is laminar. Separation then comes too late and CD too low, as if the body were already past the drag crisis. For those shapes, use a transition model or a scale-resolving model such as LES. An LES needs a fine mesh and short time steps, and the Simcenter STAR-CCM+ documentation has guidelines for it under Scale-Resolving Simulations.
Force Coefficient and Frontal Area reports
Simcenter STAR-CCM+ computes the coefficients with a Force Coefficient report. It sums the pressure and shear forces on the parts you select, takes the component along a Direction vector and divides by ½ρrefvref²aref [14]. You make one report for drag, with the direction of the free stream, and a copy for lift, with the direction normal to it. For the reference area you create a Frontal Area report, which projects the selected parts onto a plane normal to a direction you set. Since it uses rasterisation, the area has a small uncertainty of its own [14]. The wetted surface area, which a Surface Integral report of a field function equal to 1 gives you, is the wrong reference for CD on a bluff body.
The report does not check its reference values against the inflow. Reference Density, Reference Velocity and Reference Area only scale the force, and the coefficient is only as right as what you type there [14].
Say that you simulate the sphere from the example above, but the Reference Velocity in the report still says 1 m/s from an earlier case while the inlet runs at 10 m/s. The report then divides the 0.226 N by q = ½ × 1.225 × 1² = 0.61 Pa instead of 61.25 Pa and shows CD = 47.0, which you notice at once. A Reference Velocity of 12 m/s from another run is harder to catch, since the report then shows 0.326, a value that looks reasonable for a sphere. The STAR-CCM+ check in the calculator compares the coefficient in your report with the one from the force and the actual inflow, and if you also type the report’s reference values, it names the one that differs.
Worked example: a drag force of 0.226 newtons divided by the dynamic pressure, 61.25 pascals, and by the reference area, 0.007854 square metres, gives the drag coefficient 0.470, which has no unit. In the Simcenter STAR-CCM+ Force Coefficient report the force comes from Parts, Direction and Force Option, the dynamic pressure from Reference Density and Reference Velocity, and the area from Reference Area.
Angle of attack, half models and 2D
When you turn the inflow instead of the body to set an angle of attack α in the x–y plane, you need to turn both directions with it. With the flow along x at α = 0, the drag direction becomes (cos α, sin α, 0) and the lift direction (−sin α, cos α, 0). The Coordinate System property only changes how the Direction is read, and vector results are always in laboratory coordinates [14].
On a closed surface the Reference Pressure only affects the accuracy, and the documentation suggests a value near the average surface pressure. On an open surface, such as a body standing on the ground or a wing that ends at a wall, it acts as the pressure on the back side and changes the force directly [14].
For a half model on a symmetry plane, the report gives the force on the half you meshed. Use the half-model area as the Reference Area, or double the force and use the full-body area [14].
A 2D simulation gives forces per unit thickness. An Extrusion Thickness on the region’s idealization changes that only in reports with Account for Idealization switched on [14]. You then use the reference length times 1 m as the reference area, for example the diameter of the cylinder, and the result is cd or cl per unit span. In the calculator, switch to 2D section.
For a pitching moment, the Moment Coefficient report adds an Axis, an Axis Origin and a Reference Radius, which is the reference length lref in CM = M/(qAlref) [14].
In an unsteady run such as the vortex street, CD and CL oscillate. Average them over a whole number of shedding cycles, for example with a Statistics report set to Mean over the last N samples (how to set up a sliding sample window). To follow the coefficients while the solver runs, select both reports and choose Create Monitor and Plot from Report [14].
If a coefficient from your own simulation does not look right, use the form below and choose Technical question.
References and changelog
References
[1] F. M. White, Fluid Mechanics, 3rd ed., McGraw-Hill, 1994, Table 7.2 (drag of two-dimensional bodies at Re ≥ 10⁴) and Table 7.3 (drag of three-dimensional bodies at Re ≥ 10⁴).
[2] NASA Glenn Research Center, Shape effects on drag, Beginner’s Guide to Aeronautics, updated 9 September 2025.
[3] NASA Glenn Research Center, Drag equation, Beginner’s Guide to Aeronautics, updated 1 July 2025.
[4] NASA Glenn Research Center, Lift equation, Beginner’s Guide to Aeronautics, updated 10 July 2024.
[5] R. Clift, J. R. Grace and M. E. Weber, Bubbles, Drops, and Particles, Academic Press, 1978, Table 5.2 (standard drag curve for a rigid sphere).
[6] E. Achenbach, Experiments on the flow past spheres at very high Reynolds numbers, Journal of Fluid Mechanics 54(3), 565–575, 1972.
[7] E. Achenbach, The effects of surface roughness and tunnel blockage on the flow past spheres, Journal of Fluid Mechanics 65(1), 113–125, 1974.
[8] J. Li, M. Tsubokura and M. Tsunoda, Numerical investigation of drag crisis of a sphere and the effect of surface roughness, 10th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM10), Marbella, 2014.
[9] J. D. Anderson, Fundamentals of Aerodynamics, 6th ed., McGraw-Hill Education, 2017, chapter 4 (thin airfoil theory), chapter 5 (lifting-line theory) and chapter 8 (stagnation conditions).
[10] C. L. Ladson, Effects of independent variation of Mach and Reynolds numbers on the low-speed aerodynamic characteristics of the NACA 0012 airfoil section, NASA TM-4074, 1988 (tabulated data as distributed by the NASA Turbulence Modeling Resource).
[11] G. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums, Transactions of the Cambridge Philosophical Society 9, part II, 8–106, 1851.
[12] M. Behr, D. Hastreiter, S. Mittal and T. E. Tezduyar, Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries, Computer Methods in Applied Mechanics and Engineering 123, 309–316, 1995.
[13] C. H. K. Williamson, Vortex dynamics in the cylinder wake, Annual Review of Fluid Mechanics 28, 477–539, 1996.
[14] Simcenter STAR-CCM+ documentation, version 2606, Siemens Digital Industries Software, 2026 (Force Coefficient, Frontal Area, Force and Moment Coefficient reports, reporting lift and drag coefficients, idealizations).
[15] ISO 2533:1975, Standard Atmosphere, International Organization for Standardization.
[16] ITTC Recommended Procedures and Guidelines 7.5-02-01-03, Fresh water and seawater properties, International Towing Tank Conference, 2011, Table 1.
[17] D. Sucker and H. Brauer, Fluiddynamik bei quer angeströmten Zylindern, Wärme- und Stoffübertragung 8(3), 149–158, 1975; formula and discussion of the drag crisis as given by H. A. Elghannay and Y. M. F. El Hasadi, Al-Mukhtar Journal of Engineering Research 7(1), 20–28, 2023, eq. (4) and p. 21.
[18] I. H. Abbott and A. E. von Doenhoff, Theory of Wing Sections, Dover, 1959, section 6.4 (NACA four-digit sections).
Review
Reviewed by Christoffer Johansson, M.Sc., Senior Application Specialist at Volupe, on 28 October 2026.
Changelog
March 2024: first published by Christoffer Johansson.
October 2026: calculator for drag, lift and terminal velocity added, table values checked against White [1] and NASA Glenn [2], and rows without a primary source removed.
Related tools
All CFD Handbook tools →k–ω SST · k–ε · S–A
Interactive guide
Turbulence model selector
Choose a turbulence model and wall treatment for an external flow in Simcenter STAR-CCM+, with the y+ band and the checks that go with it.
Open the selector →
Re = ρUL / μ
Calculator
Dimensionless numbers
Reynolds, Mach and other numbers for your flow, to check the regime before you take a CD from a table.
Open calculator →
y+ = uτy / ν
Calculator
y+ calculator
Size the first cell for a target y+, then the prism layers around the body.
Open calculator →
GCI = Fs|ε| / (rp − 1)
Calculator
Grid Convergence Index (GCI) calculator
Estimate how much your drag and lift coefficients still depend on the mesh, from three grids.
Open calculator →
Questions about drag and lift
Page reviewed by
Christoffer Johansson
Senior Application Specialist, M.Sc.
Questions about drag or lift simulations? We reply within one day.
Ask a technical question →
How do I compare the drag of two cars?
Compare the drag area, the drag coefficient times the frontal area, since that is what sets the force, and only compare values measured the same way. A car with a coefficient of 0.30 and 2.2 m² of frontal area has about 312 N of drag at 100 km/h in air at 15 °C and needs about 8.7 kW to overcome it.
Why does the drag coefficient change with the Reynolds number?
Because the flow around the body changes. At low Reynolds numbers viscous forces dominate and the drag coefficient of a sphere falls as 24/Re. Between 10⁴ and about 2 × 10⁵ the laminar boundary layer separates near the widest point, and the coefficient stays between about 0.4 and 0.5, around White’s 0.47. At about 3.7 × 10⁵ for a smooth sphere the boundary layer turns turbulent before it separates, and the coefficient drops below 0.1 (the drag crisis). Shapes with sharp edges separate at the edges, and their coefficient hardly changes above 10⁴.
Which reference area should I use?
The one used by the data you compare with. For bluff bodies and vehicles that is the frontal area, for wings the planform (wing) area, and for a 2D section a length per unit span. A coefficient on another area converts by the ratio of the areas. For a sphere the wetted area is four times the frontal area, and its coefficient on the wetted area is a quarter of 0.47.
How do I report the drag coefficient in Simcenter STAR-CCM+?
Create a Frontal Area report for the projected area of the body. Then create a Force Coefficient report on the body’s parts, with the drag direction along the free stream and the Reference Density, Reference Velocity and Reference Area of your case. Copy it and set the direction normal to the free stream for lift. The report does not check the reference values against the inflow, and a wrong Reference Velocity gives a wrong coefficient without a warning.
Can Volupe help me simulate drag and lift in Simcenter STAR-CCM+?
Yes. Our application engineers help Simcenter STAR-CCM+ users with external aerodynamics, for example the domain, the prism layers, the turbulence model and the force reports. Customers with maintenance get this through our support, and we also run Simcenter STAR-CCM+ training courses.
What is the drag coefficient of a sphere?
About 0.47, for a smooth sphere between Reynolds numbers of 10⁴ and about 2 × 10⁵, based on its frontal area. Below a Reynolds number of about 0.1 it follows Stokes’ law, 24/Re, and above about 3.7 × 10⁵ it drops below 0.1 (the drag crisis).
What is the difference between the drag and lift coefficient?
Both are a force divided by the dynamic pressure and a reference area. The drag coefficient uses the force along the free stream, the lift coefficient the force perpendicular to it. A sphere without spin has drag but, on average, no lift. A wing is shaped for a high lift coefficient at a low drag coefficient, and the ratio of the two is its lift-to-drag ratio L/D.
Talk to an engineer in your region
Every Volupe business developer holds an engineering degree. Find your regional contact below, or send us a message.
Ask us about Drag coefficient calculator
Use this form for licence quotes, demos, training and technical questions.
What happens next
- A reply within one day.
- A scoping call with one of our engineers.
- A quote, demo or other next step that fits your enquiry.
After purchase: support from Volupe simulation engineers.