Added Mass Coefficients Potential Flow

Added Mass Coefficients, Potential Flow

(aerodynamics/aeroelasticity/added-mass-coefficients-potential-flow)

Use when you must determine the added mass coefficient (virtual mass, apparent mass) of a body accelerating rectilinearly through an inviscid irrotational fluid, from the kinetic energy of the irrotational flow the body sets up. An accelerating body must drag a surrounding mass of fluid with it, and that fluid inertia is a real load: the catalog below gives the coefficient m_a per shape from T = (1/2)rhooint_S phi*(dphi/dn) dS (Lamb Hydrodynamics Art. 136 form, normal derivative positive into the body), which for translation at speed U equals (1/2)m_aU^2. This leaf is the fluid-inertia counterpart of the unsteady siblings of this pack: aeroelastic-gust-response and flutter-speed-prediction both document neglecting apparent-mass terms at their level, and the coefficient an accelerating body carries is exactly the content they disclaim. Method provenance is the classical literature (Lamb, Hydrodynamics, Art. 136 kinetic-energy integrals and Art. 114 ellipsoid coefficients; Brennen, "A Review of Added Mass and Fluid Inertial Forces"), cited by name only, with NACA Report 824 as reference-only context for the thin-airfoil potential-flow methodology this catalog complements.

Domain quick reference

• Kinetic-energy definition: for a body translating at speed U along a principal direction, T = (1/2)rhooint_S phi*(dphi/dn) dS = (1/2)*m_a *U^2 fixes the added mass m_a of the shape. The anchor evaluates the integral on the exact dipole potentials of the moving circle and sphere and recovers the catalog coefficients below.
• Displaced fluid mass and coefficient ratio: M_disp = (4/3)rhopiab*c for an ellipsoid of semi-axes a, b, c; k = m_a/M_disp. Sphere k = 1/2 exactly; flat bodies can exceed unity (an oblate disk at rest in its plane carries zero, but pushed edge-on it drags more than its own displaced mass).
• 2-D catalog per unit span (kg/m): circular cylinder rhopiR^2 (any in-plane direction); normal flat plate of half-width a (rhopia^2 normal, EXACTLY 0.0 tangential); elliptic cylinder x^2/a^2 + y^2/b^2 = 1 with rhopib^2 along a and rhopia^2 along b (motion along one semi-axis couples to the other squared), containing the circle and plate limits.
• 3-D catalog (kg): sphere (2/3)rhopiR^3, exactly half its displaced mass; prolate and oblate spheroids via the elementary reduction of the Lamb ellipsoid coefficients alpha0, beta0, gamma0 (which sum to 2), with the translation added mass along a principal semi-axis equal to M_dispalpha/(2 - alpha) for the coefficient alpha of that axis.
• Prolate spheroid (axial a >= equatorial b = c), e = sqrt(1 - (b/a)^2): alpha0 = 2*(1 - e^2)/e^3*(0.5*ln((1+e)/(1-e)) - e), beta0 = gamma0 = 1
  • alpha0/2. Axial m_a = M_dispalpha0/(2 - alpha0), transverse m_a = M_dispbeta0/(2 - beta0). Limits: sphere e = 0 gives 2/3 and k = 1/2; a needle e -> 1 gives axial k -> 0, transverse k -> 1.
• Oblate spheroid (equatorial a = b >= polar c), e = sqrt(1 - (c/a)^2): gamma0 = 2*(e - (c/a)asin(e))/e^3, alpha0 = beta0 = 1 - gamma0/2. Polar m_a = M_dispgamma0/(2 - gamma0), equatorial m_a = M_disp*alpha0/ (2 - alpha0). Limits: sphere e = 0 gives 2/3; a thin disk c -> 0 gives gamma0 -> 2 and polar m_a -> the finite classical (8/3)rhoa^3 while the equatorial coefficient vanishes.
• Energy and reaction relations at a translation speed U and an acceleration: fluid kinetic energy T = (1/2)m_aU^2 (kinetic energy of translation); acceleration-reaction force F = m_a*acceleration the body must supply, with the opposing reaction on the body its negative; virtual mass m_body + m_a; added-mass fraction m_a/(m_body + m_a). Simultaneous orthogonal translation components superpose in the kinetic energy T = (1/2)sum(m_iU_i^2).
• Units are SI: rho in kg/m^3, lengths in m, speeds in m/s, accelerations in m/s^2; 2-D results per unit span in kg/m, 3-D results in kg. Motion is rectilinear along a principal direction; the flow is fully attached and irrotational by construction.

Workflow

1. Fix the translation problem: the shape, the principal direction of motion (which semi-axis), the fluid density rho, the translation speed U and the acceleration, and the body mass m_body for the inertia model. Note whether the result must be per unit span (2-D section) or total (3-D body).
2. Catalog coefficient lookup: select the catalog function of the shape (cylinder_added_mass, flat_plate_added_masses, elliptic_cylinder_added_masses, sphere_added_mass, prolate_spheroid_added_masses or oblate_spheroid_added_masses), call it at (rho, dimensions), and report m_a plus the ratio k = m_a/M_disp against the displaced fluid mass (4/3)rhopiab*c of the shape. The sphere branches a = b and a = c are handled exactly with no division by the eccentricity.
3. Energy picture: close the energy balance with kinetic_energy_of_translation(m_a, speed), the fluid kinetic energy (1/2)m_aU^2 the translation speed carries; compare it against the work the launch or impact must supply.
4. Inertia model: build the effective mass of the accelerating body with virtual_mass(m_body, m_added) and added_mass_fraction(m_added, m_body); in water the fluid share is typically the larger half.
5. Acceleration-reaction check: size the reaction the body must supply with acceleration_reaction_force(m_added, acceleration), whose Newton third-law partner on the body opposes the acceleration.
6. Deterministic offline check: confirm the catalog values, the anchors and the ValueError rejections with the contract test scripts/test_added_mass_coefficients_potential_flow.py.

Worked example

Two fluids exercise the catalog: water at rho = 1000 kg/m^3 (fresh-water nominal, the seaplane ditching and underwater housing context) and air at rho_air = 1.225 kg/m^3 (the airship hull). All values are the real outputs of the module.

• Surface-integral derivation (R = 1 m, rho = 1000, U = 2 m/s): the translating-circle dipole potential gives T = 6283.185307 J/m from the kinetic-energy surface integral, equal to (1/2)m_aU^2 with m_a = rhopiR^2 = 3141.592654 kg/m; the translating-sphere doublet gives T = 4188.790205 J with m_a = (2/3)rhopi*R^3 = 2094.395102 kg. The catalog coefficients ARE the integrated kinetic-energy result.
• 2-D catalog per unit span, rho = 1000 kg/m^3: circular float cylinder R = 0.5 m, 785.398163 kg/m; ditching flat plate of half-width a = 0.75 m (1.5 m wide), normal 1767.145868 kg/m and tangential 0.000000 kg/m exactly; elliptic float section a = 1.0 m, b = 0.5 m, m_along_a = 785.398163 kg/m and m_along_b = 3141.592654 kg/m; the circle limit a = b = 0.5 m returns 785.398163 kg/m both ways, difference from the cylinder 0.000e+00.
• 3-D catalog: sphere R = 0.5 m, rho = 1000, 261.799388 kg, exactly half the displaced mass (ratio 1.000000000). Oblate underwater housing a = 1.0 m, c = 0.35 m: displaced fluid mass 1466.076572 kg; polar m_a = 2422.924052 kg with k = 1.652659 ABOVE unity (the flat dome carries more fluid inertia than the mass of the fluid it displaces) and equatorial m_a = 340.526959 kg with k = 0.232271. Airship hull as a prolate spheroid a = 35 m, b = 7 m (70 m hull, 14 m diameter) in air: displaced air mass 8800.124621 kg; axial m_a = 520.273672 kg (k = 0.059121, under 6 percent of the displaced air mass, why a conventional aircraft can neglect it) and transverse m_a = 7869.604193 kg (k = 0.894261, which a 15.3-times-slender-than-a-sphere hull cannot neglect).
• Energy and reaction relations (rho = 1000, sphere R = 0.5 m, m_a = 261.799388 kg): fluid kinetic energy at U = 4 m/s, 2094.395102 J; acceleration reaction to sustain 4 m/s^2, 1047.197551 N; virtual mass on a 200 kg body, 461.799388 kg with added-mass fraction 0.566912 (the fluid inertia is the larger half of the virtual mass in water). The airship hull on a 6000 kg structure has an axial virtual mass of 6520.273672 kg and an axial added-mass fraction of 0.079793; the axial acceleration reaction at 0.2 m/s^2 is 104.054734 N versus a transverse 1573.920839 N, a 15.1 times gap any pitch or heave inertia model must carry.
• Read-off: the plate normal coefficient 1767.145868 kg/m is the per-unit-span inertia the ditching impact load sees while the tangential 0.0 confirms the plate slides without disturbing the flow; k_polar = 1.652659 of the oblate housing shows flattening toward a disk RAISES the polar coefficient toward the (8/3)rhoa^3 thin-disk asymptote (the module reproduces 0.999970 of it at c/a = 1e-4) while the equatorial coefficient collapses to 0; the airship axial/transverse k pair 0.059121/0.894261 brackets the sphere's 0.5 exactly as the slender-body limits predict (at a/b = 10 the pair is 0.020706/0.960235).

Verification

• Confirm the catalog entries of the worked example: the module returns cylinder 785.398163 kg/m at R = 0.5 m, plate (1767.145868, 0.0) kg/m at a = 0.75 m with the tangential entry EXACTLY 0.0, sphere 261.799388 kg at R = 0.5 m, oblate polar 2422.924052 kg and the airship axial 520.273672 kg, all within the tolerances asserted in the contract test.
• Confirm the sphere branches: prolate_spheroid_added_masses(rho, R, R) and oblate_spheroid_added_masses(rho, R, R) equal sphere_added_mass(rho, R) on both entries, difference 0.000e+00.
• Confirm the Lamb reductions: the closed-form alpha0/beta0/gamma0 match the independently quadrature-pinned rows (prolate a/b = 2, 5, 10: alpha0 0.347127995, 0.111641940, 0.040571761; oblate a/c = 2, 5, 10: gamma0 1.054400565, 1.500967825, 1.721608553) within 1e-9 and obey alpha0 + 2beta0 = 2 and gamma0 + 2alpha0 = 2 on every row.
• Confirm the limit branches: alpha0 at a/b = 1 + 1e-6 is 0.666666133 and gamma0 at a/c = 1 + 1e-6 is 0.666667200, each within 1e-5 of 2/3; gamma0 at c/a = 1e-4 is 1.999685881; the slender prolate k pair at a/b = 10 is 0.020706/0.960235.
• Confirm every non-physical input raises ValueError: rho 0 and negative rho on every shape function; R 0 on the cylinder and sphere; a 0 on the flat plate; b 0 on the elliptic cylinder; b 0 and b > a on the prolate spheroid; c 0 and c > a on the oblate spheroid; m_added 0 on the four scalar relations; m_body 0 on the virtual mass and the added-mass fraction; a negative speed on the kinetic energy.
• Confirm determinism: repeat runs reproduce the coefficients exactly; the module imports only math and never seeds an RNG.

Related leaves

• aerodynamics/aeroelasticity/aeroelastic-gust-response: dynamic gust response of the flexible typical section with Wagner and Kussner lag-state lift; it documents neglecting apparent-mass terms at its level, the gap this catalog fills.
• aerodynamics/aeroelasticity/flutter-speed-prediction: the V-g flutter sweep with the complex lift-deficiency function, where fluid inertia appears only as embedded terms of the oscillatory section loads.
• aerodynamics/aeroelasticity/divergence-speed: the static torsional divergence condition, no fluid inertia content.
• aerodynamics/airfoil/thin-airfoil-section-theory: steady thin-airfoil section loads, the steady counterpart of the flow this catalog integrates.
• aerodynamics/cfd/panel-method: steady 3-D doublet and source panel solutions; it holds the tag potential-flow for steady work.
• structures/loads/gust-maneuver-loads: the rigid discrete-gust load factor of the certification method; it applies gust velocities, not body acceleration inertia.

Pitfalls

• Importing a steady-flow coefficient for an accelerating body: steady panel or airfoil loads carry no kinetic-energy term; the added mass of the accelerating body must be added separately, and in water it is usually the larger half of the virtual mass (fraction 0.566912 on the worked sphere).
• Mixing the displaced fluid mass with the added mass: k = m_a/M_disp is 1/2 for a sphere but 1.652659 for the worked oblate housing, so quoting M_disp as the inertia of a flat body understates the reaction by more than 60 percent.
• Assigning a tangential coefficient to a plate: the normal flat plate moving in its own plane disturbs no irrotational flow, so its tangential coefficient is EXACTLY 0.0.
• Swapping the spheroid axes: the prolate call needs a >= b (axial semi-axis first) and the oblate call a >= c; the guards raise ValueError on b > a and c > a instead of silently evaluating the wrong shape.
• Reaching beyond the catalog: oscillatory thin-airfoil load coefficients, the complex C(k) machinery, indicial gust states, viscous or compressible or free-surface effects, rotating-body added moments and general triaxial ellipsoids (whose Lamb integrals are elliptic) are all outside this leaf.
• Reporting per-unit-span values as totals: the 2-D cylinder, plate and elliptic cylinder results are in kg/m per unit span, the sphere and spheroid results in kg; a 2-D coefficient must be multiplied by the span to enter a 3-D inertia model.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_added_mass_coefficients_potential_flow.py

35 tests: the kinetic-energy worked example and the two surface-integral anchors, the acceleration-reaction worked example and the airship axial/transverse reaction gap, the virtual mass and added-mass fraction of the worked sphere and airship, the full 2-D and 3-D catalog entries with their limits (circle, plate and sphere branches, the thin-disk asymptote at 0.999970 of (8/3)rhoa^3, the slender a/b = 10 k pair), the closed-form Lamb coefficients against the quadrature-pinned rows and their sum rules, sphere-limit continuity, ValueError rejection of every non-physical input, and repeat-run determinism. The test also passes under the pre-push hook interpreter (~/.pyenv/versions/3.13.12).

Compliance

• compliance: STANDARDS-REF, gated: false.
• Standards referenced, not reproduced: NACA Report 824 (Summary of Airfoil Data, public domain) is reference-only context for the thin-airfoil potential-flow methodology this catalog complements; no standard text appears anywhere in this leaf.
• Method provenance is the classical literature by name only: Lamb, Hydrodynamics, Art. 136 (kinetic-energy surface integrals) and Art. 114 (ellipsoid coefficients), and Brennen, "A Review of Added Mass and Fluid Inertial Forces" (1982) for the engineering catalog class.