Eulerian Derivations of Non-Inertial Navier-Stokes Equations

EULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS

ML Combrinck∗ , LN Dala∗∗ ∗Flamengro, a div of Armscor SOC Ltd & University of Pretoria, ∗∗Council of Scientiﬁc and Industrial Research & University of Pretoria

Keywords: Galilean Transformation, Rotational Frame, Compressible Flow

Abstract These accelerations are multiplied by the density to obtain the momentum form of the fictitious ef- The paper presents an Eulerian derivation of the fects and included on the right hand side of the non-inertial Navier-Stokes equations as an alter- momentum equation. native to the Lagrangian fluid parcel approach. This approach, although simple and intuitive, This work expands on the work of Kageyama and is not rigorous and lends itself to mistakes with Hyodo [1] who derived the incompressible mo- regards to the nature of the fictitious forces. It has mentum equation for constant rotation for geo- been observed in literature that these fictitious ef- physical applications. In this paper the deriva- fects are erroneously added in the conservation of tion is done for the Navier-Stokes equations in energy equation when this Lagrangian approach compressible flow for arbitrary rotation for im- is used. plementation in aero-ballistic applications. Kageyama and Hyodo [1] proposed an Eu- lerian method for the derivation of the Coriolis 1 Introduction and centrifugal forces in the momentum equation. The derivation was limited to incompress- Derivation of the non-inertial Navier-Stokes ible flow in pure rotation since their application equations (conservation of mass, momentum and is in the Geophysical field. energy) is generally done using the fluid parcel In this paper the work of Kageyama and Hy- approach. Although this method leads to the cor- odo [1] is expanded upon to include the full set of rect set of equations, it does not clearly indicate Navier-Stokes equations for compressible flow in the origin of the fictitious forces and can lead to arbitrary rotation for aero-ballistic applications. misconceptions. In deriving the conservation of momentum equation, Newton’s second law is modified to in- 2 Non-inertial Navier-Stokes Equations for clude the fictitious forces as body forces in the Constant, Pure Rotation same manner as which the gravity force is han- dled: This section involves the derivation of the non- X X inertial Navier-Stokes equations for constant ro- F + Ffictitious = ma (1) tation. This section is based on the work of Kageyama & Hyodo [1] and forms the basis for The fictitious forces are derived separately using a point mass method to obtain a relation the subsequent section where the work is ex- for the inertial acceleration in terms of the non- panded upon to derive the full set of Navier- inertial acceleration components [2]: Stokes equations for compressible flow in variable rotation. d2X a = +Ω˙ ∧x+Ω∧(Ω∧x)+V˙ +2Ω∧V (2) dt2

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2.1 Frame Transformations a constant vector of motion. In Figure 2 such a motion is described between frame O and O’. The ﬁrst step in the derivation is to deﬁne the re- Assume that the origins of the two frames in- lation of the inertial and non-inertial frames with tersect at time t = 0 and that frame O’ is moving respect to each other. These relations are math- at a constant velocity V in the x-direction. At ematically described in terms of transformation time t =∆t, the frames O and O’ are then dis- operators and is used to change the perspective tance xrel from each other. of the observer. Assume that three (3) frames exist; O, O’ and Fig. 2 Galilean Transformation between Frames Ô as indicated in Figure 1. Frame O is the stationary, inertial frame. Frame O’ is an orientation preserving frame(i and i’ has the same orientation), which can be either inertial or non- inertial depending on the cases analysed. This frame shares an origin with the rotational frame Ô. Frame Ô is the non-inertial, rotational frame The relationship between the co-ordinates and is therefore not orientation preserving. points for this single event between frames O and Now consider a point P which can be ob- O’ is described by Equation 3. This is known as served from all the frames. Point P is rotating the standard Galilean transform. around the origin of frame O, but it is stationary x0 = x − V ∆t in frames O’ and Ô. The set of equations will be y0 = y developed to describe the motion of point P in the 0 (3) rotational frame Ô. z = z t0 = t

Fig. 1 Frame Transformations Lets further assume at this point that the constant motion need not be in the x-direction alone and that it can be presented as a vector of motion as shown in Figure 3. Lets further assume that it can be used to described constant motion in rotation as well.

Fig. 3 Modiﬁed Galilean Transformation between Frames This point is described in frame O from where a modiﬁed Galilean transformation, GM, will be used to describe it in frame O’. The rotational transform, RΩt, will then be used to transform the resulting equations (as described in frame O’) to the rotational frame Ô.

2.1.1 Modiﬁed Galilean Transformation In order to simplify this case let all the frames The standard Galilean transform is limited in share the same origin and let the point P be sta- its application to constant translation in the x- tionary in the rotational frame Ô. Therefore point direction. Kageyama & Hyodo [1] modiﬁed it P is rotating with a constant angular velocity around the origin or the inertial frame O. The x to accommodate constant rotational conditions. rel component can then be described as: The Galilean transform is used to transform between two reference frames that only differ by xrel = V∆t (4)

2 EULERIAN DERIVATION OF NON-INERTIAL NAVIER-STOKES EQUATIONS where same manner is the second and third columns the projection of y0 and z0 respectively on the V = Ω ∧ x (5) rotational axes. The quantitative values of the The modified Galilean transform operator is in- rotation tensor will be different for each case. troduced such that any vector observed from the inertial frame O can be related to the vector ob- Now that the modified Galilean invariance served from the orientation preserving frame O’ and the rotational transform has been described as: for constant rotational conditions, both can be used in the derivation of the non-inertial 0 0 M u (x , t) = G u(x, t) (6) Navier-Stokes equations for constant rotation. This definition will lead to a mathematical description to directly relate the vector fields in the 2.2 Incompressible Flow Conditions inertial frame O, to the vector fields in the orientation preserving frame O’: In this section the non-inertial Navier-Stokes equations for conservation of mass, momentum u0(x0, t) = GMu(x, t) and energy for constant rotation in incompress- = GΩ∧xu(x, t) ible flow will be derived using an Eulerian ap- (7) proach. = u(x, t) − Ω ∧ x u0(x0, t) = u(x, t) + x ∧ Ω 2.2.1 Continuity Equation 2.1.2 Rotational Transformation The conservation of mass, known as the continu- Since frame Ô shares an origin with the frame ity equation, in the inertial frames takes the form: O’ the vector components in Ô is related to O’ Ωt ∂ρ by defining a rotational transform, R . Equation + (∇ · ρu) = 0 (11) 7 can be used to describe a vector as seen from ∂t frame Ô in relation to a vector in frame O. The first term represents the temporal change in û(ˆx, t) = RΩtu0(x0, t) density due to compressibility of the flow. Since Ωt Ω∧x (8) this case involves incompressible flow this term = R G u(x, t) can be neglected, but for the purposes of the derivation it will remain in the equation until RΩt is therefore the rotational transform that op- the last step. The second term is the divergence erates on x0 to obtain the ˆx co-ordinates in the of density and velocity which represents the rotational frame. From Equation 7 and 8 it can residual mass flux of a given control volume. be derived that for the velocity vector the following relation holds: Scalars, such as density and mass flux, are û(ˆx, t) = RΩt{u(x, t) + x ∧ Ω} (9) invariant under Galilean transformation. The first term of the continuity equation in the inertial Lets assume, for convenience sake, that the rota- frame can therefore be directly equated to the tion is around the z-axis of frame O. The vector term in the non-inertial frame: Ω is then described as Ω = (0, 0, Ω). The rota- ∂ρˆ ∂ρ tional transform in this case will be described by = RΩt (12) the following tensor: ∂t ∂t

 cos Ωt sin Ωt 0 The second term of the continuity equation will Ωt be affected by both frame transformations since R = − sin Ωt cos Ωt 0 (10) 0 0 1 it contains the velocity vector:

The ﬁrst column of this tensor is the projection (∇ˆ · ρˆuˆ) = RΩtGΩ∧x(∇ · ρu) (13) of the x0 component on ˆx,ˆy and ˆz. In the = RΩt[∇ · ρ(GΩ∧xu)]

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Equation 9 is used to complete the modified Galilean transformation, and the equation be- The first term that will be transformed to obtain comes: an expression that relates the inertial to the rotational frame is the time dependant term. It will be (∇ˆ · ρûˆ) = RΩt{∇ · ρ(u + x ∧ Ω)} (14) done by finding an expression for the time deriva- = RΩt{∇ · (ρu) + ∇ · ρ(x ∧ Ω)} tive in the limit:

The second term of the equation above falls away ∂û û(ˆxt+∆t, t + ∆t) − û(ˆx, t) (ˆxt, t) = lim (21) since the divergence of the cross product of dis- ∂t ∆t→0 ∆t tance and rotation is zero: An expression for û(ˆxt+∆t, t + ∆t) must be ∇ · (x ∧ Ω) = 0 (15) found. The form that the expression must take, The equation thus becomes: will directly relate the frames to each other:

∇ˆ · ρûˆ = RΩt(∇ · ρu) (16) û(ˆx , t + ∆t) = RΩ(t+∆t)GΩ∧xt+∆t t+∆t (22) [u(xt+∆t, t + ∆t)] The addition of Equation 12 and Equation 16 The tools that is required to obtain an expression leads to a relation between the continuity equa- for the relation above is described in the derivation in the inertial and rotational frames: tion below. ∂ρˆ ∂ρ + ∇ˆ · ρûˆ = RΩt( + ∇ · ρu) (17) ∂t ∂t Perform a Taylor series expansion for xt+∆t: The right hand side of the equation is equal to 2 zero since this represents the continuity equation xt+∆t = xt + ∆tV + O(∆t ) (23) in the inertial frame (Equation 11): The resulting series is truncated at the second or- ∂ρˆ + ∇ˆ · ρûˆ = 0 (18) der term and the derivative term is substituted ∂t through Equation 5. Re-arrangement of the terms Since this is the incompressible case, the tempo- will lead to an expression for displacement over ral term is equal to zero. The continuity equation the specific time interval: for the rotational frame therefore takes the form: xt+∆t − xt = x∆t = ∆t(Ω ∧ x) (24) ∇ˆ · ρûˆ = 0 (19) A Fourier series expansion is done for The physical meaning of this equation describes u(xt+∆t, t + ∆t), and with substitution of the very nature of incompressible flow assump- Equation 24 it results in: tion; the residual mass flux in a specific control volume is zero. This means that there are no u(xt+∆t, t + ∆t) compressible effects in the flow because the same = u(xt, t) + [∆t(Ω ∧ x) · ∇]u(xt, t) (25) amount of mass flux that enters a domain will exit ∂ + (∆t )u(x , t) it. ∂t t The transient density term causes a change in the residual mass flux in the domain that mani- Equation 25 is substituted into Equation 22 to get fests itself in the form of compressibility. the expression:

2.2.2 Momentum Equation ˆu(ˆxt+∆t, t + ∆t)

Ω(t+∆t) Ω∧xt+∆t The inertial equation for incompressible momen- = R G {u(xt, t) (26) ∂ tum conservation is describe by the equation be- + [∆t(Ω ∧ x) · ∇]u(x , t) + (∆t )u(x , t)} low: t ∂t t

∂u 2 + (u · ∇)u = −∇p + ν∇ u (20) GΩ∧xt+∆t can be simpliﬁed as shown below if ∂t

4 EULERIAN DERIVATION OF NON-INERTIAL NAVIER-STOKES EQUATIONS

Equation 23 is substituted in the operator and By using Equation 32, and after re-arrangement truncated at the ﬁrst order: of the terms the following expression is arrived 2 GΩ∧xt+∆t = GΩ∧{xt+∆t(Ω∧xt+O[∆t ])} at:

Ω∧{xt(1+O[∆t])} ∂û Ωt ∂ = G (27) (ˆxt, t) = R [ + (Ω ∧ x) · ∇ − Ω∧] ∂t ∂t (34) Ω∧x ≈ G t Ω∧x [G u(xt, t)] The expression for û(ˆxt+∆t, t + ∆t), then be- Substitute Equation 9 into the equation above comes: will result in: û(ˆxt+∆t, t + ∆t) ∂û Ω(t+∆t) Ω∧x (ˆxt, t) = R G {u(xt, t) ∂t + [∆t(Ω ∧ x) · ∇]u(x , t) (28) Ωt ∂ t = R [ + (Ω ∧ x) · ∇ − Ω∧](u(xt, t)) ∂ ∂t (35) + (∆t )u(x , t)} ∂ ∂t t + RΩt[ + (Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) ∂t A final set of tools is required before the expressions for Equation 21 can be completed. In the equation above the transient component of ∂ The assumption was made that point P is [ ∂t + (Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) is equal to zero: fixed in the rotating frame and the rotation is ∂ ∂x ∂Ω around the shared origin or the frames, then an (x ∧ Ω) = ∧ Ω + (x ∧ ) = 0 (36) ∂t ∂t ∂t expression can be derived for xt: Ω(t+∆t) Ωt The first term is zero because the magnitude of x ˆx = R xt+∆t = R xt (29) is constant over the time domain; its magnitude Ω∆t xt = R xt+∆t does not change with respect to the origin since This relation is substituted in the Taylor series ex- this case involves pure rotation. The second term pansion for x : is zero due to constant rotation of the point P. In t+∆t the case where the rotation is not constant, this 2 xt+∆t = xt + ∆tV + O[∆t ] term will play a role as seen in this next section. Ω∆t 2 (30) = R xt+∆t + ∆t(Ω ∧ xt) + O[∆t ] By introduction of the identity below, the Re-arrange this equation and consider in the limit terms [(Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) can be as ∆t approaches 0: simplified: Ω∆t R xt+∆t − xt+∆t lim = lim (xt ∧ Ω) (31) (a · ∇)(x ∧ Ω) = a ∧ Ω (37) ∆t→0 ∆t ∆t→0 Considering this relation for any vector b, and The entire term is hence cancelled out: take into account that xt+∆t → xt as ∆t → 0, the [(Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) following equation is arrived at: = [(Ω ∧ x) · ∇](x ∧ Ω) − Ω ∧ (x ∧ Ω) (38) RΩ∆tb − b lim = b ∧ Ω (32) = Ω ∧ (x ∧ Ω) − Ω ∧ (x ∧ Ω) ∆t→0 ∆t = 0 This leads to the final description of the unsteady Substitute Equation 28 into Equation 21 to terms in the momentum equation. Note the ap- obtain the equation: pearance of one part of the Coriolis effect mani- ∂û festing in the relation below. (ˆxt, t) ∂t ∂û Ω(t+∆t) Ω∧x 1 (ˆxt, t) R G {[1 − Ω∆t ∂t R (33) = lim ∂ ∆t→0 ∆t = RΩt[ + (Ω ∧ x) · ∇ − Ω∧](u(x , t)) (39) ∂ ∂t t +(∆t(Ω ∧ x) · ∇)]u(xt, t) + (∆t ∂t )u(xt, t)} ∆t

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The relation of the inertial to the rotational advec- If it is considered that: tion term is described in the following manner: ∇2(x ∧ Ω) = 0 (46) (û · ∇ˆ )û It can be shown that the diffusion term is invariant = RΩtGΩ∧x(u · ∇)u (40) under constant transformation: Ωt Ω∧x Ω∧x = R (G u · ∇)G u ν∇ˆ2û = RΩtν∇2u (47) Substitution of Equation 9 into the equation Note that the pressure and viscous terms are above results in: Galilean invariant in this instance and combine (û · ∇ˆ )û the two components in a vector f(x, t): = RΩt[(u + x ∧ Ω) · ∇](u + x ∧ Ω) f(x, t) = −∇p + ν∇2u (48) (41) Ωt = R [(u + x ∧ Ω) · ∇]u The new, combined parameter in the inertial and + RΩt[(u + x ∧ Ω) · ∇](x ∧ Ω) rotational frames is related in the following manner due to the invariance: Dividing out all the terms gives the final relation of the advection term between the frames. Note ˆf(ˆx, t) = RΩtf(x, t) (49) the appearance of the centrifugal effect and the other part of the Coriolis effect from the transfor- The transformation of the momentum is com- mation of the advection term. pleted through the summation of the unsteady (û · ∇ˆ )û and advection terms in the rotational and inertial = RΩt[(u · ∇)u + ((x ∧ Ω) · ∇)u (42) frames as determined in Equation 39 and Equa- tion 42: + (u ∧ Ω) + (x ∧ Ω) ∧ Ω] ∂û + (û · ∇ˆ )û The gradient of pressure term in the momentum ∂t ∂u equation is described between the frames in the = RΩt[ + (u · ∇)u + 2u ∧ Ω following manner: ∂t + x ∧ Ω ∧ Ω] (50) ˆ Ωt Ω∧x (43) ∇pˆ = R G ∇p ∂u = RΩt[ + (u · ∇)u] It was discussed earlier that scalars are invari- ∂t ant under the modified Galilean transformation. + RΩt[2u ∧ Ω + x ∧ Ω ∧ Ω] Scalars will not be invariant under the rotational The first term grouping of the equation above is transform if spatial operations is performed on it simplified as shown in the equations below. This since the axis along which the discretization is was done using Equation 20, Equation 48 and performed, changes between frames. The rela- Equation 49. tion between the gradient of pressure in the iner- ∂u tial and rotational frames is therefore described RΩt[ + (u · ∇)u] by: ∂t = RΩt[−∆p + ν∇2u] ∇ˆ pˆ = RΩt∇p (44) (51) = RΩtf(x, t) ˆ The diffusion term in the inertial frame can be = f(ˆx, t) related to the rotational frame in the following The second term grouping, with the insertion of manner: Equation 9, becomes: ν∇ˆ2û RΩt[2u ∧ Ω + x ∧ Ω ∧ Ω] = RΩtGΩ∧xν∇2u = 2(RΩtu) ∧ Ω + (RΩtx) ∧ Ω ∧ Ω = RΩtν∇2GΩ∧xu (45) = 2[û − RΩt(x ∧ Ω)] ∧ Ω (52) = RΩtν∇2(u + x ∧ Ω) + (RΩtx) ∧ Ω ∧ Ω = RΩtν[∇2u + ∇2(x ∧ Ω)] = 2û ∧ Ω − ˆx ∧ Ω ∧ Ω

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The two simplifications above is filled back into by the normal pressure forces is transform be- Equation 50 and results in the non-inertial mo- tween the frames and Equation 9 is inserted: mentum equation for constant rotation. − pˆ(∇ˆ · û) ∂û 2 + (û · ∇ˆ )û = −∇ˆ pˆ + ν∇ˆ û = RΩtGΩ∧x[−p(∇ · u)] ∂t (53) (58) + 2û ∧ Ω − ˆx ∧ Ω ∧ Ω = RΩt[−p∇ · (u + x ∧ Ω)] = RΩt[−p∇ · u + −p∇ · (x ∧ Ω)] It can be seen from the equation above that the fictitious forces associated with constant rotation It can be shown that this terms is also invariant is the centrifugal and the Coriolis effects. The under transformation by the insertion of Equation centrifugal effect originates from the transforma- 15: tion of the advection terms while the Coriolis effect is form both the transient and advection −pˆ(∇ˆ · û) = RΩt[−p∇ · u] (59) terms.

2.2.3 Energy Equation The diffusion is invariant under transformation since the heat transfer coefficient (k) and temper- The general energy equation in the inertial frame ature (T) are scalars. The transformation between takes the following form: the frames then becomes: ∂ρε + (∇ · ρεu) = −p(∇ · u) + ∇ · (k∇T ) (54) ∇ˆ · (kˆ∇ˆ Tˆ) = RΩtGΩ∧x[∇ · (k∇T )] ∂t (60) = RΩt[∇ · (k∇T )] The time dependant term is transformed in a similar manner to the pressure term in the momentum equation since internal energy is a scalar. It All the transformed terms of the energy equation was already discussed that scalars are invariant is summed to obtain the equation below. under transformation. The first term is therefore ∂ρˆεˆ transformed though the following operation: + (∇ˆ · ρˆεˆû) +p ˆ(∇ˆ · û) − ∇ˆ · (kˆ∇ˆ Tˆ) ∂t ∂ρˆεˆ ∂ρε ∂ρε = RΩt (55) = RΩt[ + (∇ · ρεu) + p(∇ · u) (61) ∂t ∂t ∂t − ∇ · (k∇T )] The convective term is transformed between the frames with the use of the rotational transform, The right hand side of the equation is equal to modified Galilean transform and by substitution zero, as shown in Equation 54. The energy equa- of Equation 9 tion in the non-inertial frame for constant rotation is invariant under transformation in this specific (∇ˆ · ρˆεˆû) case: Ωt Ω∧x = R G (∇ · ρεu) ∂ρˆεˆ (56) + (∇ˆ · ρˆεˆû) = −pˆ(∇ˆ · û) + ∇ˆ · (kˆ∇ˆ Tˆ) (62) = RΩt[∇ · ρε(u + x ∧ Ω)] ∂t = RΩt[∇ · ρεu + ∇ · ρε(x ∧ Ω)] This equation can be further simplified with the It was already shown in Equation 15 that the sec- assumption of incompressibility and using Equa- ond term on the right hand side is equal to zero. tion 19: Therefore the transformed equation becomes: ∂ρˆεˆ + (∇ˆ · ρˆεˆû) = ∇ˆ · (kˆ∇ˆ Tˆ) (63) (∇ˆ · ρˆεˆû) = RΩt(∇ · ρεu) (57) ∂t Note that the energy equation is invariant under The terms that represents the rate of work done transformation.

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2.3 Compressible Flow Conditions 34 (which defined the relation between the incompressible transient term in different frames) In this section the non-inertial Navier-Stokes the above becomes: equations for conservation of mass, momentum ∂ ∂u and energy for constant rotation in compressible (ˆρû) = RΩtGΩ∧x[ρ + ρ(Ω ∧ x) · ∇u ∂t ∂t (67) flow will be derived using an Eulerian approach. ∂ρ − ρΩ ∧ u + u ] ∂t 2.3.1 Continuity Equation The product rule is then used to combine the The general continuity equation in the rotational ∂u ∂ρ terms ρ ∂t and u ∂t so that the equation simpli- frame was derived in Equation 18. This has fies to: shown that the equation is invariant under trans- ∂ ∂ formation. The compressible, non-inertial equa- (ˆρû) = RΩt[ (ρ) + ρ(Ω ∧ x) · ∇ tion thus remains: ∂t ∂t (68) − ρΩ∧]GΩ∧xu ∂ρˆ + ∇ˆ · ρûˆ = 0 (64) ∂t The equation is of the same form as Equation 34, 2.3.2 Momentum Equation it can therefore be shown that the final form of the equation will be similar to Equation 39, but The incompressible form of the momentum equa- with the inclusion of the density scalar: tion as shown in Equation 20, made the as- ∂ ∂ sumption that the change in density is negligible. (ˆρû) = RΩt[ (ρ) + ρ(Ω ∧ x) · ∇ Therefore, the equation could be simplified by di- ∂t ∂t (69) viding density into all the terms as there are no − ρΩ∧]u temporal or spatial gradients in density. The diffusion term in particular could be simplified in a The relation between the frames for the advection manner that would facilitate easy transformation term in the compressible Navier-Stokes momen- where the divergence of the gradient of velocity tum equation is: yields the same result as taking the laplacian of ˆ Ωt Ω∧x the velocity. This is not the case when compress- ∇ · (ˆρû ⊗ û) = R G [∇ · (ρu ⊗ u)] (70) ibility has to be accounted for. The compressible By using Equation 7 the equation above is ex- Navier-Stokes Equation in the inertial frame will panded into: take the form: ˆ ∂ ∇ · (ˆρû ⊗ û) ρu + ∇ · (ρu ⊗ u) ∂t (65) = RΩt{∇ · ρ[(u + x ∧ Ω) ⊗ (u + x ∧ Ω)]} T = −∇P + ∇ · [µ(∇u + ∇u ) + λ(∇ · u)I] = RΩt{(∇ · ρu) ⊗ u + (∇ · ρu) ⊗ (x ∧ Ω) (71) The non-inertial form of the separate terms of + [∇ · ρ(x ∧ Ω)] ⊗ u the equation, must be derived from this form to + [∇ · ρ(x ∧ Ω)] ⊗ (x ∧ Ω)} obtain the compressible transformation. As shown in the previous section, the identity be- First consider the unsteady term in the rota- low can be used to simplify the equation. tional frame and apply the product rule for (∇ · a) ⊗ (x ∧ Ω) = a ∧ Ω (72) partial derivatives. This operation will result in two terms that was not considered during the This will lead to the following expression for re- incompressible case: lating the diffusion term in the rotational frame to ∂ ∂û ∂ρˆ the terms in the inertial frame: (ˆρû) =ρ ˆ + û (66) ∂t ∂t ∂t ∇ˆ · (ˆρû ⊗ û) With the aid of Equations 8 (which defined the = RΩt[∇ · ρu ⊗ u + ρu ∧ Ω (73) transformation between the frames) and Equation + (∇ · ρ(x ∧ Ω) ⊗ u + (ρx ∧ Ω) ∧ Ω]

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It must be noted that the incompressible momen- The pressure gradient term in the momentum tum equation is a special case of the compress- equation is transformed in a similar manner than ible momentum equation where it is assumed that shown in the previous section. This part of the the velocity of the flow is low enough (generally equation remain invariant since it is a scalar. incompressibility assumptions is assumed below Mach 0.3) that the compressible effects does not ∇ˆ Pˆ = RΩtGΩ∧x∇P (74) have a significant effect on the flow properties. Ωt ∇ˆ Pˆ = R ∇P No truly incompressible conditions exist.

In the transformation of the diffusion term the 2.3.3 Energy Equation difference between the compressible and incompressible cases must be noted. The divergence of The general energy equation remains the same as the velocity vector is not equal to zero, therefore described in the previous section: the completed diffusion term must be accounted ∂ρε for. The expression for relating the diffusion term + (∇ · ρεu) = −p(∇ · u) + ∇ · (k∇T ) (81) ∂t between the frames hence becomes: This equation remains invariant in the non- ∇ˆ · [ˆµ(∇ˆ û + ∇ˆ ûT ) + λˆ(∇ˆ · û)Î] inertial frame as shown in Equation 62 Ωt Ω∧x T = R G ∇ · [µ(∇u + ∇u ) (75) ∂ρˆεˆ + (∇ˆ · ρˆεˆû) = −pˆ(∇ˆ · û) + ∇ˆ · (kˆ∇ˆ Tˆ) (82) + λ(∇ · u)I] ∂t With the implementation of Equation 9, the right Take again note that there are not fictitious effects hand side of the equations becomes: present in the energy equation.

RΩt∇ · {µ[∇(u + x ∧ Ω) + ∇(u + x ∧ Ω)T ] 2.4 Incompressible Equations as a Special (76) + λ(∇ · (u + x ∧ Ω))I} Case of the Compressible Equations

If it is considered that, The assumption of incompressibility can be made when the flow velocity is substantially ∇(x ∧ Ω) + ∇(x ∧ Ω)T = 0 (77) small (below Mach 0.3) so that the mass flux is close to zero. This is a special case of compress- and ible flow that assumes that no temporal changes in density occurs. As such, if the compressible ∇ · (x ∧ Ω) = 0 (78) Navier-Stokes equations where derived correctly, applying the incompressible assumptions to It can be shown that, as in the case of incom- it, should lead to the derived, incompressible pressible flow, the diffusion component of the Navier-Stokes equations. momentum equation is invariant for constant rotation conditions: The compressible continuity equation in the rotational frame was determined to be: ∇ˆ · [ˆµ(∇ˆ û + ∇ˆ ûT ) + λˆ(∇ˆ · û)Î] (79) ∂ρˆ = RΩt∇ · [µ(∇u + ∇uT ) + λ(∇ · u)I] + ∇ˆ · ρûˆ = 0 (83) ∂t The same principle as in the previous section is The applied assumption of incompressibility will used to derive the final equation: result in the transient change in density being zero: ∂ρˆû ˆ ∂ρˆ + ∇ · (ˆρû ⊗ û) = 0 (84) ∂t ∂t = −∇ˆ Pˆ (80) The equation therefore becomes: + ∇ˆ · [ˆµ(∇ˆ û + ∇ˆ ûT ) + λˆ(∇ˆ · û)Î] + 2ρû ∧ Ω − ρˆx ∧ Ω ∧ Ω ∇ˆ · ρûˆ = 0 (85)

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This provides the same result Equation 19 to the non-inertial momentum equation: which is the derived, incompressible continuity ∂û equation in the rotational frame. + (û · ∇ˆ )û = −∇ˆ pˆ + ν∇ˆ 2û ∂t (91) The derived, compressible momentum equation + 2û ∧ Ω − ˆx ∧ Ω ∧ Ω in the rotational frame is: This equation above it the same as Equation 53 ∂ρˆû which was derived from first principles. + ∇ˆ · (ˆρû ⊗ û) ∂t ˆ ˆ = −∇P (86) The conservation of energy equation in the + ∇ˆ · [ˆµ(∇ˆ û + ∇ˆ ûT ) + λˆ(∇ˆ · û)Î] rotational frame for compressible flow is described by: + 2ρû ∧ Ω − ρˆx ∧ Ω ∧ Ω ∂ρˆεˆ + (∇ˆ · ρˆεˆû) = −pˆ(∇ˆ · û) + ∇ˆ · (kˆ∇ˆ Tˆ) (92) The first term that must be simplified to account ∂t from incompressibility is the diffusion term: If the continuity equation is applied to this equa- ∇ˆ · [ˆµ(∇ˆ û + ∇ˆ ûT ) + λˆ(∇ˆ · û)Î] (87) tion it will result in: ∂ρˆεˆ Lets consider the x-momentum components of + (∇ˆ · ρˆεˆû) = ∇ˆ · (kˆ∇ˆ Tˆ) (93) the divergence of the deviatoric stress tensor. As- ∂t sume in this instance that the dynamic viscosity This is the same as Equation 63 where the µ is a constant. Simplify the relation below to incompressible energy equation in the rotational obtain the form as shown below: frame was derived. ∂ ∂u ∂ ∂u ∂v (2µ + λ∇ · u) + (µ( + )) ∂x ∂t ∂y ∂y ∂x This section therefore indicates that there ∂ ∂u ∂w are no observed discrepancies between the + (µ( + )) ∂z ∂z ∂x derived equations for the compressible and ∂2u ∂2u ∂2u ∂2v ∂2w incompressible cases in the rotational frame. =µ(2 + + + + ) (88) ∂x2 ∂y2 ∂z2 ∂y∂x ∂z∂x ∂ 3 Non-inertial Navier-Stokes Equations for + (λ∇ · u) ∂x Variable, Pure Rotation ∂ ∂ = µ∇2u + µ (∇ · u) + (λ∇ · u) ∂x ∂x In this section the non-inertial Navier-Stokes This relation can be written in the vector form to equation for variable rotation around the axis of account for all the components of the diffusive the inertial frame will be derived. momentum if it is assumed that the second vis- It will observed that the equations for mass cosity, λ, is constant (Stokes Hypothesis): and energy conservation for constant and variable rotation remains the same. The equation for con- ∇ˆ · µˆ∇ˆ û + ∇ˆ · [(ˆµ + λˆ)(∇ˆ · û)] (89) servation of momentum has an additional term to account for the variability in rotation. It will be The second term in the relation above will be shown that the equations for constant and vari- equal to zero if the incompressible continuity able rotational acceleration are equivalent and equation is substituted into the relation. This re- that all additional terms due to the changes in sult in the following equation: acceleration is negligible in the limit as ∆t ap- ∂ρˆû proaches zero. + ∇ˆ · (ˆρû ⊗ û) = −∇ˆ Pˆ + ∇ˆ · µˆ∇ˆ û ∂t (90) + 2ρû ∧ Ω − ρˆx ∧ Ω ∧ Ω 3.1 Frame Transformations

Since density is constant in the equation above, it Assume that the same three frame exist as in the can be divided into the equation, which will lead previous derivation: O, O’ and Ô. O is again the

10 EULERIAN DERIVATION OF NON-INERTIAL NAVIER-STOKES EQUATIONS stationary frame, O’ is the orientation preserving In this equation it can already be seen that the frame and Ô is the rotational frame. In the previ- effect of further derivatives on xrel becomes ous derivation Ô was rotating at a constant veloc- smaller and smaller. Further on in this section ity around the shared origin. In this instance Ô it will be shown that only one additional term are rotating around the shared origin with a con- are introduced in the non-inertial Navier-Stokes stant rotational acceleration. momentum equation due to variable acceleration. The continuity and energy conservation 3.1.1 Modified Galilean Transformation equations remains invariant. In this section the modified Galilean transform In the same manner as in the previous sec- obtained from the previous section will be aug- tion, Equation 7, the relation between the mented to account for rotational acceleration. order preserving frame and the inertial frame is Assume that the frame origins intersect at defined with the inclusion of the accelerating time t = 0 and that frame O’ is moving at ac- components: celeration a in three dimensional space. At time u0(x0, t) t =∆t frame O and O’ are distance xrel from M each other. In Equation 4 there was not an ac- = G u(x, t) ˙ (99) celerating component, but in this case it must be = GΩ∧x+(Ω∧x)∆tu(x, t) incorporated in the expression to account for the = u(x, t) + x ∧ Ω + (x ∧ Ω˙ )∆t distance travelled by the particle: 1 x = V∆t + a∆t2 (94) 3.1.2 Rotational Transformation rel 2 The rotational transform for this case can be de- In the equation above the velocity is again defined in the same manner as Equation 8. The vec- scribed as in Equation 5: tor components in Ô is related to O’ by defining a rotational transform and substituting Equation V = Ω ∧ x (95) 99 to relate Ô to O: ˙ 2 The acceleration is the time derivative of the ve- û(ˆx, t) = RΩt+Ωt u0(x0, t) (100) locity: ˙ 2 ˙ = RΩt+Ωt GΩ∧x+(Ω∧x)∆tu(x, t)

∂V ∂ ˙ 2 = Ω ∧ x RΩt+Ωt is the rotational transform that operates ∂t ∂t (96) on x0 to obtain the ˆx co-ordinates in the acceler- ∂Ω ∂x = ∧ x + Ω ∧ ating, rotational frame. ∂t ∂t The second term is equal to zero since this case Lets assume, for convenience sake, that the involves pure rotation. The accelerating compo- rotation is around the z-axis, then the vector nent for the rotational case is therefore expressed Ω is described as Ω = (0, 0, Ω) and vector Ω˙ as: is described as Ω˙ = (0, 0, Ω)˙ . The rotational transform in this case will be described by: a = Ω˙ ∧ x (97)  cos(Ωt + Ω˙ t2) sin(Ωt + Ω˙ t2) 0 Ωt+Ω˙ t2 R = − sin(Ωt + Ω˙ t2) cos(Ωt + Ω˙ t2) 0 Equation 94 is a Taylor series expansion that was 0 0 1 truncated after the second order term since con- (101) stant acceleration was assumed. Had the acceleration not been constant, the additional terms From Equation 99 and 100 it can be derived will be accounted for by the inclusion of further that for the velocity vector the following relation derivative terms: holds: ˆu(ˆx, t) 1 1 (102) x = V∆t + a∆t2 + ˙a∆t3 + ... (98) Ωt+Ω˙ t2 ˙ rel 2! 3! = R [u(x, t) + x ∧ Ω + (x ∧ Ω)∆t]

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3.2 Incompressible Flow Conditions With the help of Equation 103 and the assumption that the flow is incompressible, the final In this section the non-inertial Navier-Stokes equation for mass conservation in the accelerat- equations for conservation of mass, momentum ing frame is obtained: and energy for variable rotation in incompressible flow will be derived using an Eulerian ap- ∇ˆ · ρûˆ = 0 (110) proach. Consider the term (x ∧ Ω˙ )∆t in Equation 106. 3.2.1 Continuity Equation This term originates from the Taylor series expansion in Equation 98 and contains a ∆t compo- Consider the continuity equation in the inertial nent. Any further expansions, due to changes in reference frame: acceleration, will also contain a ∆tn component. ∂ρ These acceleration terms will become negligible + (∇ · ρu) = 0 (103) ∂t in the limit:

As was discussed in the previous section, scalars lim (x ∧ Ω˙ )∆t = 0 (111) are invariant under transformation and therefore ∆t→0 the time dependant term in the inertial and accel- In the light of this, the derivations for the conti- erating frame is related by: nuity equation in the rotational frame have been

∂ρˆ ˙ 2 ∂ρ proven to be invariant to rotation whether the ac- = RΩt+Ωt (104) ∂t ∂t celeration it is zero, constant or variable. In any rotational frame Equation 110 holds for incom- The relation of the second term in the continuity pressible conditions. equation becomes: 3.2.2 Momentum Equation (∇ˆ · ρûˆ) The conservation of momentum equation in the ˙ 2 ˙ = RΩt+Ωt GΩ∧x+(Ω∧x)∆t(∇ · ρu) (105) inertial frame can be expressed by: Ωt+Ω˙ t2 Ω∧x+(Ω˙ ∧x)∆t = R ∇ · ρ(G u)} ∂u + (u · ∇)u = −∇p + ν∇2u (112) With the implementation of Equation 102 the re- ∂t lation is simplified to: The terms will again, as in the previous section, be treated separately and then combined to (∇ˆ · ρûˆ) (106) obtain the final transformed equation. = RΩt∇ · ρ[u + x ∧ Ω + (x ∧ Ω˙ )∆t] The first transformation will concern the The second and third terms in the relation above unsteady term where an expression must be is equal to zero: found for: ∇ · (x ∧ Ω) = 0 ∂û (107) (ˆxt, t) ∇ · (x ∧ Ω˙ ) = 0 ∂t û(ˆx , t + ∆t) − û(ˆx, t) = lim t+∆t (113) The relation is hence simplified to an invariant ∆t→0 ∆t relation:

Ωt+Ω˙ t2 ∇ˆ · ρûˆ = R (∇ · ρu) (108) The first task is to find an expression for the term û(ˆxt+∆t, t + ∆t). The expression must take the The addition of Equation 104 and Equation 108 form: gives a relation for continuity in the rotational û(ˆxt+∆t, t + ∆t) frame: ˙ 2 2 = RΩ(t+∆t)+Ω(t +∆t ) (114) ∂ρˆ Ωt+Ω˙ t2 ∂ρ ˙ + ∇ˆ · ρûˆ = R ( + ∇ · ρu) (109) Ω∧xt+∆t+(Ω∧xt+∆t)∆t ∂t ∂t [G u(xt+∆t, t + ∆t)]

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For simplification the rotational and modified In order to complete the expression in Equation Galilean transforms will be shown in the follow- 113 further expressions must be defined. Assume ing manner: that the point P is fixed in the rotating frame, and ˙ 2 2 t+∆t the rotation is around the origin (meaning that L RΩ(t+∆t)+Ω(t +∆t ) = RM (115) and L’ share an origin), then: ˙ t+∆t GΩ∧xt+∆t+(Ω∧xt+∆t)∆t = GM M t+∆t M t ˆx = R xt+∆t = R xt The Taylor series expansion for x is ex- (122) t+∆t M ∆t pressed as: xt = R xt+∆t

1 2 3 Use the expression above and conduct a Taylor xt+∆t = xt + ∆tV + ∆t a + O(∆t ) (116) 2 series expansion for xt+∆t: The equation above is truncate at the second or- 1 2 3 der. With the substitution of Equation 96 and xt+∆t = xt + ∆tV + ∆t a + O[∆t ] 2 Equation 97 and further re-arrangement the equa- M ∆t tion becomes: = R xt+∆t + ∆t(Ω ∧ xt) (123) 1 x − x + ∆t2(Ω˙ ∧ x ) + O[∆t3] t+∆t t 2 t 1 (117) = x = ∆t(Ω ∧ x) + ∆t2(Ω˙ ∧ x) ∆t 2 Re-arrange the expression above and consider it The Fourier series expansion is obtained for in the limit: ∆t u(xt+∆t, t + ∆t). Substitute Equation 117 into RM x − x lim t+∆t t+∆t the expression to obtain: ∆t→0 ∆t

u(xt+∆t, t + ∆t) = lim [(xt ∧ Ω) (124) ∆t→0 = u(xt, t) + {[∆t(Ω ∧ x) 1 3 − ∆t(Ω˙ ∧ xt) − O[∆t ]] 1 2 (118) 2 + ∆t (Ω˙ ∧ x)] · ∇}u(xt, t) 2 If the above is considered for any vector b, and ∂ + (∆t )u(xt, t) if it is taken into account that xt+∆t → xt as ∂t ∆t → 0, the following equation related to ro- The equation above is substituted into Equation tation is obtained: 114 to get the expression: RM ∆t b − b û(ˆxt+∆t, t + ∆t) lim = b ∧ Ω (125) ∆t→0 ∆t t+∆t t+∆t = RM GM {u(x , t) + [∆t(Ω ∧ x) t With all the required expressions in place Equa- 1 (119) + ∆t2(Ω˙ ∧ x)] · ∇]u(x , t) tion 113 can now be completed: 2 t ∂ RM t+∆t GM t {[1 − 1 + (∆t )u(x , t)} ∂û RΩ∆t t (ˆxt, t) = lim ∂t ∂t ∆t→0 ∆t In the previous section is was shown that the fol- +∆t(Ω ∧ x) · ∇ lowing simplification can be made: ∆t (126) M t+∆t M t 1 2 ˙ G ≈ G (120) + 2 ∆t (Ω ∧ x) · ∇]u(xt, t) ∆t Using the simplification for the modified +(∆t ∂ )u(x , t)} Galilean transformation above leads to the ex- ∂t t pression: ∆t û(ˆx , t + ∆t) Equation 125 is used to simplify the expression t+∆t above and with some re-arrangement of terms the M t+∆t M t = R G {u(xt, t) following expression is obtained: 1 2 ˙ (121) + [∆t(Ω ∧ x) + ∆t (Ω ∧ x)] · ∇]u(xt, t) ∂û M t ∂ 2 (ˆxt, t) = R [ + (Ω ∧ x) · ∇ − Ω∧] ∂ ∂t ∂t (127) + (∆t )u(x , t)} M t ∂t t [G u(xt, t)]

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This equation above will retain its current form of terms in Equation 128, will now be considered. irrespective of any further changes in acceleration. All other terms that in inserted to account The transient component of the terms can for variation in acceleration will become negligi- be expanded again using the product rule. In this ble when the expression is considered in the limit. case the terms are all equal to zero:

∂ ∂x ∂Ω˙ Equation 102 is substituted in the equation (x ∧ Ω˙ ) = ∧ Ω˙ + (x ∧ ) = 0 (132) above to remove the modified Galilean operator ∂t ∂t ∂t from the equation: The first term in the equation above is equal to ∂û zero because the magnitude of x is constant. The (ˆx , t) ∂t t second term in the equation above is equal to t ∂ zero because constant acceleration is considered = RM [ + (Ω ∧ x) · ∇ − Ω∧](u(x , t)) ∂t t in this case. In the case where the acceleration t ∂ is not constant, the second term will not be zero. + RM [ + (Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) (128) ∂t However, this entire term will fall away in the t ∂ limit as ∆t, in the main equation, tends to zero. + RM [ + (Ω ∧ x) · ∇ − Ω∧](x ∧ Ω˙ )∆t ∂t Now consider the term [(Ω ∧ x) · ∇ − Ω∧](x ∧ Ω˙ )∆t. If the identity: The different parts of [ ∂ + (Ω ∧ x) · ∇ − ∂t ˙ ˙ Ω∧](x ∧ Ω) , which is the second combination (a · ∇)(x ∧ Ω) = a ∧ Ω (133) of terms in Equation 128, will now be considered. is considered, it can be shown that the entire term is equal to zero: The transient term in the equation above can be expanded on with the use of the product [(Ω ∧ x) · ∇ − Ω∧](x ∧ Ω˙ ) rule for partial differential equations: (134) = (Ω ∧ x) ∧ Ω˙ − Ω ∧ (x ∧ Ω˙ ) = 0 ∂ ∂x ∂Ω (x ∧ Ω) = ∧ Ω + (x ∧ ) (129) The entire third combination of terms in Equa- ∂t ∂t ∂t tion 128 falls away, if not due to the nature of the In the equation above the first term on the right rotational motion, it will fall away in the limit hand side is zero because the magnitude of x is due to the ∆t term. constant, it does not change its magnitude with respect to the origin. The second term is not The above leads to the final description of equal to zero in this case and has to be taken into the unsteady terms in the momentum equation account since it represents the unsteady rotation, for constant rotation: this is called the Euler fictitious force. ∂û M t ∂ (ˆxt, t) = R [ + (Ω ∧ x) · ∇ The terms [(Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) has ∂t ∂t (135) M t ˙ already been shown in the previous section to be − Ω∧](u(xt, t)) + R (x ∧ Ω) equal to zero: At this stage it must noted that in any pure rota- [(Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) = 0 (130) tion, this is the form the unsteady component of the equation will always take. Additional terms ∂ The relation [ ∂t +(Ω∧x)·∇−Ω∧](x∧Ω) will, that appear in rotation will become negligible in for this case, simplify to: the limit due to the ∆t. Take note in this equation of the appearance ∂ [ + (Ω ∧ x) · ∇ − Ω∧](x ∧ Ω) = x ∧ Ω˙ (131) of a part of the Coriolis effect and the Euler ∂t effect.

∂ The different parts of [ ∂t + (Ω ∧ x) · ∇ − The advection term in the non-inertial Navier- Ω∧](x ∧ Ω˙ )∆t, which is the third combination Stokes equation for constant rotation will be

14 EULERIAN DERIVATION OF NON-INERTIAL NAVIER-STOKES EQUATIONS transformed in the following paragraphs. The used to expand on the relation: relation between the inertial and rotational ˆ2 frames can be described by the equation below: ν∇ û t t = RM GM ν∇2u ˆ M t M t (141) (û · ∇)û = R G (u · ∇)u M t 2 M t (136) = R ν∇ G u M t M t M t = R (G u · ∇)G u t = RM ν∇2[u + x ∧ Ω + (x ∧ Ω˙ )∆t] With the use of Equation 102, the equation above If it is consider that: is expanded into: ∇2(x ∧ Ω) = 0 (142) 2 ˙ (û · ∇ˆ )û ∇ ((x ∧ Ω)∆t) = 0 t = RM [(u + x ∧ Ω + ((x ∧ Ω˙ )∆t) · ∇]u The advection terms of the equation becomes: M t ˙ t + R [(u + x ∧ Ω + ((x ∧ Ω)∆t) · ∇](x ∧ Ω) ν∇ˆ2û = RM ν∇2u (143) M t ˙ ˙ + R [(u + x ∧ Ω + ((x ∧ Ω)∆t) · ∇](x ∧ Ω)∆t Note that the pressure and viscous term is (137) Galilean invariant in this instance and the two components can be combined into a single vari- The equation above can be simplified by consid- able f: ering it in the limit of ∆t and using the identity in Equation 37. this will lead to the equation: f(x, t) = −∇p + ν∇2u (144) The relation for f between the inertial and rota- (û · ∇ˆ )û tional frames can therefore be described by: M t (138) = R [(u · ∇)u + ((x ∧ Ω) · ∇)u t ˆf(ˆx, t) = RM f(x, t) (145) + (u ∧ Ω) + (x ∧ Ω) ∧ Ω]

In this case a note can again be made that this Expressions have been obtained for all the parts is the form that the advection terms of the mo- of the momentum equation that relates the in- mentum equation will always taken in rotation, ertial frame to the rotational frame. The trans- irrespective if the rotation is constant or variable. formed equation is obtained by the summation Note in this equation the appearance of of the transient and advection components as de- the other part of the Coriolis effect and the rived in Equation 135 and Equation 138: centrifugal effect. ∂û + (û · ∇ˆ )û The pressure gradient term will be consid- ∂t t ∂u ered next. The relation between the inertial and = RM [ + (u · ∇)u + 2u ∧ Ω ∂t rotational frame for the pressure gradient can be ˙ (146) expressed in the manner below: + x ∧ Ω ∧ Ω + x ∧ Ω] M t ∂u t t = R [ + (u · ∇)u] ∇ˆ pˆ = RM GM ∇p (139) ∂t t + RM [2u ∧ Ω + x ∧ Ω ∧ Ω + x ∧ Ω˙ ] Since scalars are invariant under transformation the equation can be simplified to: First grouping of terms on the right hand side of the equation above can be simplified using Equa- t ∇ˆ pˆ = RM ∇p (140) tions 112, 144 and 145:

t ∂u RM [ + (u · ∇)u] ∂t t The last term in the momentum equation that = RM [−∇p + ν∇2u] must be transformed is the diffusion term. The (147) M t relation between the inertial and non-inertial = R f(x, t) frames is described below and Equation 102 is = ˆf(ˆx, t)

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Second grouping of terms in the transformed Equation 102: equation can be simplified using Equation 102 and with some manipulation the equation will be- (∇ˆ · ρˆεˆû) t t come: = RM GM (∇ · ρεu) t t M ˙ RM [2u ∧ Ω + x ∧ Ω ∧ Ω + x ∧ Ω˙ ] = R [∇ · ρε(u + x ∧ Ω + ((x ∧ Ω)∆t))] (152) (148) M t = 2û ∧ Ω − ˆx ∧ Ω ∧ Ω + ˆx ∧ Ω˙ = R [∇ · ρεu + ∇ · ρε(x ∧ Ω) + ∇ · ρε(x ∧ Ω˙ )∆t] The above simplifications can be substituted into Equation 146 and will result in the non-inertial The second and third terms on the right hand side momentum equation in a rotational frame: of the equation above was shown in Equation 107 to be equal to zero. The convective term there- ∂û + (û · ∇ˆ )û fore becomes Galilean invariant under transfor- ∂t mation: ˆ ˆ 2 (149) = −∇pˆ + ν∇ û t (∇ˆ · ρˆεˆû) = RM (∇ · ρεu) (153) + 2û ∧ Ω − ˆx ∧ Ω ∧ Ω + ˆx ∧ Ω˙

It can be noted that the only difference between The term that represents the rate of work done by the momentum equation in constant and variable the normal force can be related in the inertial and rotation is the appearance of the Euler term. rotational frames as shown below. This term can The Euler term ˆx∧Ω˙ represents the unsteady be expanded upon using Equation 102. rotational acceleration of the point P around the ˆ axis. It has also been shown that the equation − pˆ(∇ · ˆu) t t above will always take this form whether the ac- = RM GM [−p(∇ · u)] t celeration in rotation is constant or variable. In = RM [−p∇ · (u + x ∧ Ω + (x ∧ Ω˙ )∆t] (154) the case where there is no acceleration, the Euler M t term will fall away. = R [−p∇ · u − p∇ · (x ∧ Ω) − p∇ · (x ∧ Ω˙ )∆t] 3.2.3 Energy Equation Showing that the second and third terms is again Consider the energy equation in the inertial equal to zero, the same as above and indicated in frame: Equation 107, this transformation is also invariant. ∂ρε + (∇ · ρεu) = −p(∇ · u) + ∇ · (k∇T ) (150) t ∂t −pˆ(∇ˆ · ˆu) = RM [−p∇ · u] (155) The various terms can be transformed to the rotational frame separately and then combined The diffusive term in the rotational frame can be to obtained the energy equation in the rotational expressed in the inertial frame with the following frame. relation:

t t The time dependant term can be related be- ∇ˆ · (kˆ∇ˆ Tˆ) = RM GM [∇ · (k∇T )] (156) tween the frames as shown below since scalars Since k and T are scalars the relation is invariant are invariant under transformation: under transformation:

∂ρˆεˆ t ∂ρε = RM (151) ˆ ˆ ˆ ˆ M t ∂t ∂t ∇ · (k∇T ) = R [∇ · (k∇T )] (157)

The relation for the convective term between the The full relation between the rotational and in- inertial and rotational frame is shown below. This ertial frames for the energy equation can be ob- equation can be expanded upon with the used of tained by summation of the components obtained

16 EULERIAN DERIVATION OF NON-INERTIAL NAVIER-STOKES EQUATIONS above. This leads to the equation: In the case only pure rotations was considered, if ∂ρˆεˆ the rotation was not pure, an additional fictitious + (∇ˆ · ρˆεˆû) +p ˆ(∇ˆ · û) − ∇ˆ · (kˆ∇ˆ Tˆ) ∂t term would have been present in the momentum t ∂ρε equation. This term has its origin from the tran- = RM [ + (∇ · ρεu) + p(∇ · u) (158) ∂t sient term as shown in Equation 129. The result- − ∇ · (k∇T )] ing momentum equation would be: The right hand side of the equation is equal to ∂ρˆû + ∇ˆ · (ˆρû ⊗ û) zero, this can be seen from re-arrangement of the ∂t ˆ ˆ ˆ ˆ ˆ T ˆ ˆ ˆ terms in Equation 150. The non-inertial energy = −∇P + ∇ · [ˆµ(∇û + ∇û ) + λ(∇ · û)I] (162) equation is invariant under transformation in this + 2ρû ∧ Ω − ρˆx ∧ Ω ∧ Ω specific case for constant acceleration in rotation, ˙ but it can be seen that this equation will remain in + ρˆx ∧ Ω + ρ ˙x ∧ Ω this form even if the acceleration is not constant. This equation applies to all bodies in rotational accel- Equation 110 is further used to arrive at: eration. It has been shown that no further terms will ∂ρˆεˆ + (∇ˆ · ρˆεˆû) = ∇ˆ · (kˆ∇ˆ Tˆ) (159) be added to the non-inertial formulation even if the ∂t acceleration is unsteady.

3.3 Compressible Flow Conditions 3.3.3 Energy Equation In this section the non-inertial Navier-Stokes The energy equation remains invariant in the non- equations for conservation of mass, momentum inertial frame as shown in Equation 159. The non- and energy for variable rotation in compressible inertial energy equation in compressible flow there- flow will be derived using an Eulerian approach. fore becomes: ∂ρˆεˆ 3.3.1 Continuity Equation + (∇ˆ · ρˆεˆû) = −pˆ(∇ˆ · û) + ∇ˆ · (kˆ∇ˆ Tˆ) (163) ∂t The continuity equation was shown to be invari- It was shown in this paper that the energy equa- ant under the modified Galilean transformation tion is invariant under transformation in all cases for constant rotational acceleration in Equation of rotation. 109. Since the compressible case is considered here, the transformed equation becomes: 3.4 Constant Rotation Equations as a Spe- ∂ρˆ + ∇ˆ · ρûˆ = 0 (160) cial Case of the Variable Rotation Equa- ∂t tions The continuity equation has thus proven to be invariant in all instances of rotation. In the previous sections it was shown that the continuity and conservation of energy equations 3.3.2 Momentum Equation are invariant under transformation for all cases of The difference between the compressible and in- rotation. This is not unexpected since mass and compressible momentum equation was discussed energy are scalar values that is invariant, how- in the previous section. It was shown in Section ever this is an important result to show since it 2.3.2 that the difference between the incompress- is not obvious when deriving the equations us- ible and compressible case only manifests in the ing the fluid parcel approach. The derivation of diffusion term. In a similar manner the equation the momentum equation for the various cases not for variable rotation will take a form similar to only provided the appropriate fictitious forces for Equation 80 but with the inclusion in this case of each case, but it also showed from which trans- the Euler term as seen in Equation 149: formations the forces originated. ∂ρˆû + ∇ˆ · (ˆρû ⊗ û) As was shown, the derivations are directly re- ∂t liant on the Tailor series expansion of the mo- T (161) = −∇ˆ Pˆ + ∇ˆ · [ˆµ(∇ˆ û + ∇ˆ û ) + λˆ(∇ˆ · û)Î] tion of the observed point P. The constant rota- + 2ρû ∧ Ω − ρˆx ∧ Ω ∧ Ω + ρˆx ∧ Ω˙ tion case is therefore a special case of the variable

17 ML COMBRINCK, LN DALA rotation case since the order at which the Tay- term and is also present in all rotation cases. The lor series was truncated affects the fictitious force Euler force originates from the transformation of involved. If the derivations was done correctly, the transient term and is not present when the ro- the variable rotation case will lead directly to the tational speed is constant. There is a fourth term constant rotation case in constant rotation condi- that is closely linked to the Euler force, ˙x ∧ Ω, tions is applied to it. Since the conservation of which is not present in cases of pure rotation, but mass and energy equations is invariant, this will in full arbitrary, non-constant rotations. be shown using the momentum equation. It has further been shown in this paper that Consider the momentum equation for vari- changes in rotational acceleration does not intro- able, pure rotation in the rotational frame: duce further fictitious forces in the momentum ∂û equation. The four effects identified includes the + (û · ∇ˆ )û = −∇ˆ pˆ + ν∇ˆ 2û ∂t (164) full range of terms that can be added in the rota- + 2û ∧ Ω − ˆx ∧ Ω ∧ Ω + ˆx ∧ Ω˙ tional frame. The Eulerian approach does not allow for the The fictitious forces involved are the Coriolis misconceptions that can arise when using the La- force, centrifugal force and Euler force. If it is grangian approach. The method is mathemati- considered that the rotational is constant, the Eu- cally rigorous, but more so the meaning of the ler force should fall away. The equation then be- terms is clear and leads to a improved under- comes: standing of the origin of the fictitious effects in ∂û + (û · ∇ˆ )û = −∇ˆ pˆ + ν∇ˆ 2û the rotational frame. ∂t (165) + 2û ∧ Ω − ˆx ∧ Ω ∧ Ω 5 References This is the same as Equation 53 which describes [1] Kageyama A and Hyodo M. Eulerian derivation the momentum equation for constant rotation as of the Coriolis force. Geochemistry, Geophysics seen from the rotational frame. This indicates and Geosystems, Vol. 7, No. 2, pp 1-5, 2006. consistency in the derivations. [2] White FM. Viscous Fluid Flow. Third Edition, McGraw-Hill, 2006. 4 Conclusion

This paper presented an Eulerian derivation of the 6 Nomenclature non-inerital Navier-Stokes equations for com- 6.1 Super Scripts and Sub Scripts pressible flow in arbitrary rotational conditions. It was shown that the continuity equation and 0 Orientation preserving frame the energy equation is invariant under transfor- ˆ Rotational frame mation in all cases. Some instances have been rel Relative conditions observed in the literature where the fictitious ef- t Time fects were added to the energy equation due to ∆t Change in time misconception that arise when using the fluid parcel (Lagrangian) approach. This work indicates 6.2 Alphabet that no fictitious effects are present in the energy equation. a Acceleration vector In the derivation of the non-inertial momen- b Vector tum equation the origin of the fictitious forces k Heat transfer coefficient could be observed. The Coriolis force originates p Pressure per unit mass from the transformation of both the transient and t Time the adjective terms. This term is present in all u Veloctiy vector cases of rotation. The centrifugal force origi- x Distance in x-direction nates from the transformation of the advection x Position vector

18 EULERIAN DERIVATION OF NON-INERTIAL NAVIER-STOKES EQUATIONS y Distance in y-direction Identity 2 z Distance in z-direction G Galilean operator ∇ · (x ∧ Ω˙ ) I Identity matrix ∂ = [xjΩ˙ k − xkΩ˙ j]i O Frame designations ∂xi P Pressure ∂ − [xiΩ˙ k − xkΩ˙ i]j R Rotational transform operator ∂xj T Temperature ∂ + [xiΩ˙ j − xjΩ˙ i]k V Velocity in x-direction ∂xk V Velocity vector = 0 X Position vector Identity 3 6.3 Greek Letters ∇2(x ∧ Ω) ε Internal energy ∂2 ∂2 ∂2 = ( + + )[x Ω − x Ω ]i λ Second viscosity ∂x2 ∂x2 ∂x2 j k k j µ Dynamic viscosity i j k ∂2 ∂2 ∂2 ν Kinematic viscosity − ( + + )[x Ω − x Ω ]j ∂x2 ∂x2 ∂x2 i k k i ρ Density i j k Ω Rotational speed around the z-axis ∂2 ∂2 ∂2 + ( 2 + 2 + 2 )[xiΩj − xjΩi]k Ω Rotational speed vector ∂xi ∂xj ∂xk = 0 7 Acknowledgements Identity 4 The authors would like to thank Professor Akira Kageyama for his correspondence with regards to ∇2(x ∧ Ω˙ ) 2 2 2 his paper [1]. ∂ ∂ ∂ ˙ ˙ = ( 2 + 2 + 2 )[xjΩk − xkΩj]i ∂xi ∂xj ∂xk 2 2 2 8 Contact Author Email Address ∂ ∂ ∂ ˙ ˙ − ( 2 + 2 + 2 )[xiΩk − xkΩi]j ∂xi ∂xj ∂xk 2 2 2 mailto:madeleinec@ﬂamengro.co.za ∂ ∂ ∂ ˙ ˙ + ( 2 + 2 + 2 )[xiΩj − xjΩi]k ∂xi ∂xj ∂xk Appendix A - Proof of Identities = 0

Identity 1 Identity 5

∇ · (x ∧ Ω) (c · ∇)(x ∧ Ω) = c ∧ Ω ∂ = [xjΩk − xkΩj]i ∂xi LH = (c · ∇)(x ∧ Ω) ∂ − [xiΩk − xkΩi]j ∂ ∂ ∂ ∂xj = (ci + cj + ck )[(xjΩk − xkΩj)i ∂xi ∂xj ∂xk ∂ + [xiΩj − xjΩi]k − (xiΩk − xkΩi)j + (xiΩj − xjΩi)k] ∂xk = (c Ω − c Ω )i − (c Ω − c Ω )j = 0 j k k j i k k i + (ciΩj − cjΩi)k]

19 ML COMBRINCK, LN DALA

(c · ∇)(x ∧ Ω) = c ∧ Ω Identity 8 x ∧ Ω = −Ω ∧ x RH = c ∧ Ω = (c Ω − c Ω )i − (c Ω − c Ω )j j k k j i k k i LH = x ∧ Ω + (ciΩj − cjΩi)k] = (xjΩk − xkΩj)i − (xiΩk − xkΩi)j

+ (xiΩj − xjΩi)k LH = RH

Identity 6 RH = −Ω ∧ x = −[(xkΩj − xjΩk)i − (xkΩi − xiΩk)j ˙ ˙ (c · ∇)(x ∧ Ω) = c ∧ Ω + (xjΩi − xiΩj)k]

= (xjΩk − xkΩj)i − (xiΩk − xkΩi)j ˙ LH = (c · ∇)(x ∧ Ω) + (xiΩj − xjΩi)k ∂ ∂ ∂ = (ci + cj + ck )[(xjΩ˙ k − xkΩ˙ j)i ∂xi ∂xj ∂xk LH = RH − (xiΩ˙ k − xkΩ˙ i)j + (xiΩ˙ j − xjΩ˙ i)k] Copyright Statement = (cjΩ˙ k − ckΩ˙ j)i − (ciΩ˙ k − ckΩ˙ i)j ˙ ˙ + (ciΩj − cjΩi)k] The authors confirm that they, and/or their company or or- ganization, hold copyright on all of the original material RH = c ∧ Ω˙ included in this paper. The authors also confirm that they have obtained permission, from the copyright holder of any = (cjΩ˙ k − ckΩ˙ j)i − (ciΩ˙ k − ckΩ˙ i)j third party material included in this paper, to publish it as + (c Ω˙ − c Ω˙ )k] i j j i part of their paper. The authors confirm that they give permission, or have obtained permission from the copyright LH = RH holder of this paper, for the publication and distribution of this paper as part of the ICAS 2014 proceedings or as indi- Identity 7 vidual off-prints from the proceedings.

∇(x ∧ Ω) + ∇(x ∧ Ω)T = 0

∇(x ∧ Ω) =

 ∂ ∂  (xjΩk − xkΩj) ... (xjΩk − xkΩj)  ∂xi ∂xk   ∂ ∂  (−xiΩk + xkΩi) ... (−xiΩk + xkΩi) ∂xi ∂xk  ∂ ∂   (xiΩj − xjΩi) ... (xiΩj − xjΩi)  ∂xi ∂xk    0 Ωk −Ωj = −Ωk 0 Ωi  Ωj −Ωi 0 

∇(x ∧ Ω) + ∇(x ∧ Ω)T = 0

     0 Ωk −Ωj  0 −Ωk Ωj  −Ωk 0 Ωi + Ωk 0 −Ωi = 0  Ωj −Ωi 0  −Ωj Ωi 0 