LAW OF BIOT AND SAVART

For a variety of applications in aero- and hydrodynamics, the flow induced by a concen- trated distribution of vorticity (a vortex) with arbitrary orientation must be calculated. Here we consider the simple case of incompressible flow whereV,u ¼ 0. Taking the curl of the vorticity produces:

Vu ¼ V ðVuÞ ¼ VðV,uÞ V²u ¼ V²u,

where the second equality follows from an identity of vector calculus (B.3.13). The two ends of this extended equality form a Poisson equation, and its solution is the vorticity-induced portion of the fluid velocity:

uðx,tÞ ¼ 1 4p

Z

V⁰

1

jxx⁰jðV⁰uðx⁰,tÞÞd³x⁰, (5.14)

5.5. VELOCITY INDUCED BY A VORTEX FILAMENT: LAW OF BIOT AND SAVART 181

whereV⁰encloses the vorticity of interest andV⁰operates on thex⁰coordinates (see Exercise 5.8). This result can be further simplified by rewriting the integrand in (5.14):

1

jxx⁰jðV⁰uðx⁰,tÞÞ ¼ V⁰

uðx⁰,tÞ jxx⁰j V⁰

1

jxx⁰j uðx⁰,tÞ

¼ V⁰

uðx⁰,tÞ

jxx⁰j þ xx⁰ jxx⁰j³

!

uðx⁰,tÞ,

to obtain:

uðx,tÞ ¼ 1 4p

Z

V⁰

V⁰

uðx⁰,tÞ

jxx⁰j d³x⁰ þ 1 4p

Z

V⁰

uðx⁰,tÞ ðxx⁰Þ jxx⁰j³ d³x⁰:

Here the first integral is zero whenV⁰is chosen to capture a segment of the vortex, but it takes several steps to deduce this. First, rewrite the curl operation in index notation and apply Gauss’ divergence theorem:

Z

V⁰

V⁰

uðx⁰,tÞ

jxx⁰j d³x⁰ ¼ Z

V⁰

3_kij v vx⁰_i

u_jðx⁰,tÞ

jxx⁰j d³x⁰ ¼ Z

A⁰

3_kij

u_jðx⁰,tÞ jxx⁰j n_id²x⁰

¼ Z

A⁰

nuðx⁰,tÞ

jxx⁰j d²x⁰, (5.15)

whereA⁰is the surface ofV⁰andnis the outward normal onA⁰. Now chooseV⁰to be a volume aligned so that its end surfaces are locally normal tou(x⁰,t) while its curved lateral surface lies outside the concentration of vorticity as shown inFigure 5.8. For this volume, the final integral in (5.15) is zero becausenu¼0 on its end surfaces sinceu(x⁰,t) andnare parallel there, and becauseu(x⁰,t)¼0 on its lateral surface. Thus, (5.14) reduces to:

uðx,tÞ ¼ 1 4p

Z

V⁰

uðx⁰,tÞ ðxx⁰Þ

jxx⁰j³ d³x⁰: (5.16)

If an elemental vortex segment of lengthdlis considered so thatV⁰¼DA⁰dl, and the obser- vation location,x, is sufficiently distant from the vorticity concentration locationx⁰ so that (xex⁰)/jxex⁰j³is effectively constant over the vorticity concentration, then (5.16) may be simplified to:

duðx,tÞy 1 4p

Z

DA⁰

juðx⁰,tÞjeud²x⁰ðxx⁰Þ

jxx⁰j³dl ¼ Gdl

4peuðxx⁰Þ

jxx⁰j³, (5.17)

whereduis the velocity induced by the vortex segment, andGandeuare the strength and direction of the vortex segment atx⁰, respectively. This is an expression of the Biot-Savart vortex induction law.

5. VORTICITY DYNAMICS

182

5.6. VORTICITY EQUATION IN A ROTATING FRAME

A vorticity equation was derived inSection 5.4for a fluid of uniform density observed from an inertial frame of reference. Here, this equation is generalized to a rotating frame of reference and a nonbarotropic fluid. The flow, however, will be assumed nearly incom- pressible in the Boussinesq sense, so that the continuity equation is approximately V,u¼0. And, for conciseness, the comma notation for spatial derivatives (Section 2.14) is adopted.

The first step is to show thatV,u¼u_i,iis zero. From the definitionu¼Vu, we obtain u_i,ⁱ ¼ ð3_inquq,ⁿÞ,ⁱ ¼ 3_inquq,ⁿⁱ:

In the last term,3_inqis antisymmetric iniandn, whereas the derivativeuq,niis symmetric ini and n. As the contracted product of a symmetric and an antisymmetric tensor is zero, it follows that

u_i;i ¼ 0 or V,u[0: (5.18)

Hence, the vorticity field is divergence free (solenoidal), even for compressible and unsteady flows.

The continuity and momentum equations for a nearly incompressible flow in a steadily rotating coordinate system are

ui,ⁱ ¼ 0, andvui

vt þujui;jþ23_ijkUjuk ¼ 1

rp_;iþgiþnui;jj; (5.19, 5.20) where U is the angular velocity of the coordinate system and gi is the effective gravity (including centrifugal acceleration); see Section 4.7. The advective acceleration can be written as

ω ω

x – x´ u(x,t)

V´

x´

x

FIGURE 5.8 Geometry for derivation of Law of Biot and Savart. The location of the vorticity concentration or vortex isx⁰. The location of the vortex-induced velocityuisx. The volumeV⁰contains a segment of the vortex. Its flat ends are perpendicular to the vorticity in the vortex, while its curved lateral sides lie outside the vortex.

5.6. VORTICITY EQUATION IN A ROTATING FRAME 183

ujui,^j ¼ ujðu_i,^juj,ⁱÞ þujuj,ⁱ ¼ u_j3_ijku_kþ1

2ðu_jujÞ,ⁱ ¼ ðuuÞ_iþ1

2ðu²_jÞ,ⁱ, (5.21) where we have used the relation

3_ijku_k ¼ 3_ijk3_kmnun,^m ¼ ðd_imd_jned_ind_jmÞu_n,^m ¼ uj,ⁱui,^j: (5.22)

The viscous diffusion term can be written as

nu_i;jj ¼ nðu_i;ju_j;iÞ_;jþnu_j;ij ¼ n3_ijku_k;j; (5.23) where we have used (5.22) and the fact thatu_j,ij¼0 because of (5.19). Equation(5.23)says that nV²u¼ nVu, which we have used several times before (e.g., see (4.40)). Because Uu¼ uU, the Coriolis acceleration term in (5.20) can be rewritten

23_ijkUju_k ¼ 23_ijkUku_j: (5.24)

Substituting (5.21), (5.23), and (5.24) into (5.20), we obtain

vu_i=vtþ 1

2u²_j þF

,ⁱ3_ijku_jðu_kþ2UkÞ ¼ ð1=rÞp,ⁱn3_ijku_k,^j, (5.25) where we have also setg¼ VF; see (4.18).

Equation (5.25) is another form of the Navier-Stokes momentum equation, so the rotating-frame-of-reference vorticity equation is obtained by taking its curl. Sinceu_n¼3_nqiui,q, we need to operate on (5.25) by3_nqi( ),qwhich produces:

v vt

3_nqiu_i,^q

þ3_nqi 1

2u²_j þF

,^iq3_nqi3_ijk h

u_jðu_kþ2UkÞi

,^q¼ 3_nqi 1

rp,ⁱ

,^qn3_nqi3_ijku_k,^jq: (5.26) The second term on the left side vanishes on noticing that3_nqiis antisymmetric inqandi, whereas the derivativeðu²_j=2þPÞ_;iq is symmetric inqandi. The third term on the left side of (5.26) can be written as

3nqi3_ijk h

uj

u_kþ2Uk

i

;q¼ ðdnjd_qkd_nkd_qjÞh

ujðu_kþ2UkÞi

;q

¼ h

unðu_kþ2UkÞi

;kþh

ujðu_nþ2UnÞi

;j

¼ unðu_k;kþ2Uk,^kÞ u_n;kðu_kþ2UkÞ þu_jðunþ2UnÞ_;j

¼ u_nð0þ0Þ un;kðu_kþ2UkÞ þujðu_nþ2UnÞ_;j

¼ u_n;jðu_jþ2UjÞ þu_ju_n;j; (5.27)

5. VORTICITY DYNAMICS

184

where we have usedui,i¼0,u_i,i¼0 and the fact that the derivatives ofUare zero.

The first term on the right-hand side of (5.26) can be written as

3_nqi 1

rp_;i

;q

¼ 1

r3_nqip_;iqþ 1

r²3_nqir_;qp_;i

¼ 0þ 1

r²½VrVp_n; (5.28)

which involves then-component of the vector VrVp. The viscous term in (5.26) can be written as

n3_nqi3_ijku_k;jq ¼ nðd_njd_qkd_nkd_qjÞu_k;jq

¼ nu_k;nkþnu_n;jj ¼ nu_n;jj: (5.29)

If we use (5.27) through (5.29), then (5.26) becomes vu_n

vt ¼ u_n;jðu_jþ2UjÞ uju_n;jþ 1

r²½VrVp_nþnu_n;jj: Changing the free index fromntoiproduces

Du_i

Dt ¼ ðu_jþ2UjÞu_i;jþ 1

r²½VrVp_iþnu_i;jj: In vector notation this can be written:

Du

Dt ¼ ðuD2UÞ,VuD 1

r²VrVpþnV²u: (5.30)

This is thevorticity equationfor a nearly incompressible (i.e., Boussinesq) fluid observed from a frame of reference rotating at a constant rateU. Hereuanduare, respectively, the velocity and vorticity observed in this rotating frame of reference. As vorticity is defined as twice the angular velocity, 2Uis theplanetary vorticityand (uþ2U) is theabsolute vorticityof the fluid, measured in an inertial frame. In a nonrotating frame, the vorticity equation is obtained from (5.30) by settingU to zero and interpretingu andu as the absolute velocity and vorticity, respectively.

The left side of (5.30) represents the rate of change of vorticity following a fluid particle.

The last termnV²urepresents the rate of change ofudue to molecular diffusion of vorticity, in the same way thatnV²urepresents acceleration due to diffusion of velocity. The second term on the right-hand side is the rate of generation of vorticity due to baroclinicity of the flow, as discussed inSection 5.2. In a barotropic flow, density is a function of pressure alone, soVrandVpare parallel vectors. The first term on the right side of (5.30) represents vortex stretching and plays a crucial role in the dynamics of vorticity even whenU¼0.

To better understand the vortex-stretching term, consider the natural coordinate system wheresis the arc length along a vortex line, npoints away from the center of vortex-line curvature, andmlies along the second normal tos(Figure 5.9). Then,

5.6. VORTICITY EQUATION IN A ROTATING FRAME 185