ㄽᩥ                                   Ἑᕝᢏ⾡ㄽᩥ㞟,➨20ᕳ,2014ᖺ6᭶Ἑᕝࡢᒁᡤὶゎᯒ࡟࠾ࡅࡿ㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡢ
ᑟฟ࡜ࡑࡢᚲせᛶ࣭㐺⏝ᛶ
NON-EQUILIBRIUM WALL LAW FOR LOCAL FLOWS IN RIVERS ANDDISCUSSIONS ON ITS NEED AND APPLICABILITYෆ⏣㱟ᙪ1࣭⚟ᒸᤩ஧2࣭Athanasios N. PAPANICOLAOU3
Tatsuhiko UCHIDA, Shoji FUKUOKA and Athanasios N. PAPANICOLAOU
1ṇ఍ဨ ༤(ᕤ) ୰ኸ኱Ꮫ◊✲㛤Ⓨᶵᵓ෸ᩍᤵ㸦ࠛ112-8551 ᮾி㒔ᩥி༊᫓᪥1-13-27㸧
2ࣇ࢙࣮ࣟ ᕤ༤ Ph.D. ◊✲㛤Ⓨᶵᵓᩍᤵ 㸦ྠୖ㸧
3㠀఍ဨ Ph.D.

IIHR-Hydroscience and Engineering, The University of Iowa㸦100 C. Maxwell Stanley
Hydraulics Laboratory, Iowa City, IA 52242-1585, U.S.㸧

Recent advancements of computational technology and developments of numerical scheme allow us
to calculate detail flow structures in rivers with quasi-3D and 3D calculation methods. While theconventional wall law for rough bed has been used for the boundary condition on rough bed, its
applicability to those detail calculations is questionable. This paper proposed a non-equilibrium wall law
for rough bed with momentum equations for bottom vorticity and roughness layers and discussed its
applicability through the theoretical considerations of the magnitude of each terms of the governing
equations for the non-equilibrium wall law. We applied the present non-equilibrium wall law employedby the BVC method (BVC-NWL) to the real scale 3D flow in front of structures measured in the Joganji
River field experiment. The calculation results by BVC-NWL provided better agreement with measured
data than the BVC method with the equivalent wall law (BVC-EWL).
Key Words : non-equilibrium wall law, multi scale roughness concept, large scale roughness, small
scale roughness, vortex layer, roughness layer, bottom velocity computation method㸯㸬ᗎㄽ
ィ⟬ᶵᢏ⾡ࡸᩘ್ゎᯒᡭἲࡢⓎᒎ࡟ࡼࡾヲ⣽࡞ὶࢀᵓ㐀ࡸἙᗋኚືࡀ᳨ウ࡛ࡁࡿࡼ࠺࡟࡞ࡗ࡚ࡁࡓ㸬Ἑᕝὶࡢゎᯒ⢭ᗘࡣὶࢀࡢᩘ್ゎᯒἲࡔࡅ࡛࡞ࡃ᢬ᢠࡢホ౯ἲࡸ฼⏝ࡍࡿἙᕝ㸪ὥỈὶࡢほ ࢹ࣮ࢱ࡟౫Ꮡࡍࡿ㸬ὥỈὶࡸᅵ◁⛣ືࡢヲ⣽࡞ゎᯒ࡟ࡣࡇࢀࡽࢆࣂࣛࣥࢫࡼࡃ㧗ᗘ㸰㸬♟ᗋἙᕝ࡟࠾ࡅࡿ⢒ᗘࡢከ㔜ࢫࢣ࣮ࣝᛶ࡜ᚑ᮶ࡢ᢬ᢠホ౯ἲ

Ỉ㠃zs
LSR(▼㸪ἙᗋἼ㸪
ᒙ

Ἑᗋᙧ≧)

⢒ᗘᒙ

୺ィ⟬㡿ᇦ

Flow

ᗏ㠃zb

ᅗ⢒ᗘࡢከ㔜ࢫࢣ࣮ࣝᛶ࡜᢬ᢠホ౯ἲቨἲ๎SSR(◁♟)

໬ࡍࡿᚲせࡀ࠶ࡿ㸬ࡋ࠿ࡋ㸪᢬ᢠࡢホ౯࡟࠾࠸࡚᭱ࡶ
ᇶᮏ࡜࡞ࡿᗏ㠃ࡏࢇ᩿ᛂຊࡢホ౯ἲ࡟ࡘ࠸࡚ࡣ㸪࣐ࢽࣥࢢࡢ⢒ᗘಀᩘࡸᑐᩘศᕸ๎࡟ᇶ࡙ࡃ᢬ᢠ๎ࡀ⏝࠸ࡽࢀ࡚࠾ࡾ㸪㧗ᗘ࡞ゎᯒ࡟࠾ࡅࡿ㐺⏝ᛶ࡟ㄢ㢟ࡀṧࡉࢀ࡚࠸ࡿ㸬ᮏ◊✲࡛ࡣ㸪ᗏ㠃ࡢቃ⏺᮲௳ࢆホ౯ࡍࡿࡓࡵࡢ㠀ᖹ⾮⢒㠃᢬ᢠ๎ࢆᑟฟࡋ㸪ࡑࡢᚲせᛶ࡜㐺⏝ᛶ࡟ࡘ࠸࡚⌮ㄽゎᯒ࡜⌧ᆅᐇ㦂ࡢᩘ್ゎᯒ࡟ࡼࡗ࡚᫂ࡽ࠿࡟ࡍࡿ㸬㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡢ᳨ウ࡟ඛ❧ࡕ㸪♟ᗋἙᕝ࡟࠾ࡅࡿ
ᚑ᮶ࡢ⢒㠃᢬ᢠホ౯ἲ࡜ゎᯒἲࡢㄢ㢟ࢆᩚ⌮ࡍࡿ㸬♟ᗋἙᕝ࡛ࡣ㸪ᵝࠎ࡞኱ࡁࡉࡢ⢒ᗘࡀᏑᅾࡍࡿࡓࡵ㸪ᅗ࡟♧ࡍࡼ࠺࡟㸪⢒ᗘࡢከ㔜ࢫࢣ࣮ࣝᛶࢆᑟධࡋ㸪኱ࡁ࡞⢒
ᗘ(LSR)࡜ᑠࡉ࡞⢒ᗘ(SSR)ࡢ᢬ᢠࡀศࡅ࡚ホ౯ࡉࢀ࡚ࡁ
ࡓ1)㸬⢒ᗘࡢከ㔜ࢫࢣ࣮ࣝᛶࡢᑟධࡣ♟ᗋἙᕝࡢ᢬ᢠホ౯ࡔࡅ࡛࡞ࡃ㸪౛࠼ࡤἙᗋἼࢆ᭷ࡍࡿὶࢀࡢᤲὶຊࢆホ౯ࡍࡿࡓࡵ࡟㸪ἙᗋἼࡢᙧ≧᢬ᢠ࡜⾲㠃᢬ᢠࢆศ㞳ࡍࡿ- 217 -

➼࡟ࡶ⏝࠸ࡽࢀ࡚ࡁࡓ2)㸬㏻ᖖ㸪SSRࡣ⾲㠃᢬ᢠ࡜ࡋ࡚
wh wU j h(1)
 wVb 0ྲྀࡾᢅࢃࢀࡿᑠࡉ࡞⢒ᗘᙧ≧㸪LSRࡣࡑࢀࡼࡾ኱ࡁ࡞ᙧ
wt
wx j≧ࡢ⢒ᗘࢆᣦࡋ㸪ἙᗋἼࡸỈ㊰ᙧ≧࡞࡝ࡢᙧ≧ࡶྵࡲࢀࡿ㸬኱ࡁ࡞⢒ᗘࡣὶࢀሙࡢ᢬ᢠࡸ≉ᚩࢆᨭ㓄ࡍࡿࡓࡵ㸪wz
wU i h wU iU j h
 wVbubi  gh sከࡃࡢ◊✲⪅࡟ࡼࡾホ౯᪉ἲࡀ᳨ウࡉࢀ࡚ࡁࡓ㸬
wx j
wxi
wt
(2)
ᖹ㠃஧ḟඖゎᯒ࡛ࡣ㖄┤᪉ྥࡢὶࢀࡸᅽຊࡢኚ໬ࡀ⪃
w

(
'
'
)
h
W
u
u
whdp0 dpb wzb W biijij៖࡛ࡁ࡞࠸ࡓࡵ㸪Ỉἐ኱ᆺ⢒ᗘࢆホ౯ࡍࡿࡓࡵ࡟㸪⢒ᗘ




w
w
w
U
x
U
x
U
U
x
i
i
j
ᙧ≧࡟ຍ࠼㸪ᢠຊಀᩘࢆ⏝࠸ࡓ᢬ᢠ㡯ࢆᇶ♏᪉⛬ᘧ࡟௜3)ࡇࡇ࡟㸪i,j =1(x),2(y)㸪Ui㸸Ỉ῝ᖹᆒὶ㏿㸪h㸸Ỉ῝㸪ຍࡋ࡚ホ౯ࡉࢀࡿ 㸬ࡋ࠿ࡋ㸪ᢠຊಀᩘࡣ⢒ᗘࡢ㓄⨨࡟ࡼࡗ࡚ኚ໬ࡍࡿࡔࡅ࡛࡞ࡃ㸪⢒ᗘ࡟ᑐࡋ࡚Ỉ῝ࡀὸࡃ࡞wVb㸸ᗏ㠃ࡢ ᒙ࡜ᆶ┤࡞ὶ㏿㸪ubi㸸ᗏ㠃ὶ㏿㸪zs㸪zb㸸Ỉ㠃㸪ᆅ┙㧗㸪 dp㸸ᅽຊࡢ㠀㟼Ỉᅽᡂศ(dp=pUg(zs z)㸪ࡿ࡜㸪┦ᑐỈ῝࡟ࡼࡗ࡚ࡶኚ໬ࡍࡿࡓࡵ㸪ணࡵᐃࡵࡿࡇ
p㸸ࣞ࢖ࣀࣝࢬᛂຊࡢ➼᪉ᡂศ࡜ᅽຊࡢ࿴)㸪dpb㸪dp0㸸
࡜ࡣᐜ࡛᫆ࡣ࡞࠸ၥ㢟ࡀ࠶ࡿ㸬ᗏ㠃㸪Ỉ῝ᖹᆒࡢᅽຊ೫ᕪ㸪Wbi㸸ᗏ㠃ࡏࢇ᩿ᛂຊ㸪W ij㸸୕ḟඖゎᯒ࡛ࡣ㸪Nicholas4)ࡣ኱ࡁ࡞⢒ᗘࡢᙳ㡪ࢆ୙つ๎࡞Ἑᗋࡢ㉳అ࡛⾲⌧ࡋ㸪ᑠࡉ࡞⢒ᗘ࡜ศ㞳ࡋࡓ♟ᗋἙࣞ࢖ࣀࣝࢬᛂຊࡢ㠀➼᪉ᡂศ࡜ὶ㏿㖄┤ศᕸ࡟ࡼࡿỈᖹᕝࡢゎᯒἲࢆᥦ᱌ࡋ࡚࠸ࡿ㸬୕ḟඖゎᯒ࡛ࡣ㸪ᑠࡉ࡞⢒ࡏࢇ᩿ᛂຊࢸࣥࢯࣝ㸪 u i ' u j ' 㸸ὶ㏿㖄┤ศᕸ࡟ࡼࡿ㐠ືᗘ(SSR)ࡢ᢬ᢠࡣቨἲ๎࡛ࣔࢹࣝ໬ࡉࢀࡿ㸬༶ࡕ㸪⢒ᗘ㔞஺᥮㸪ui’=ui Ui㸪࡛࠶ࡾ㸪ᮏㄽᩥ࡛ࡣୖ௜ࡁࡢࣂ࣮㸪ࡢ኱ࡁࡉ࡜ὶ㏿ホ౯Ⅼࡢ㧗ࡉ ᒙࡢཌࡉ㸪ᅗ ➼ࢆ⏝኱ᩥᏐ⾲グ㸪ࡶࡋࡃࡣୗ௜ࡢࢮ࡛ࣟỈ῝ᖹᆒ್㸪ୗ௜ࡢ࠸࡚ᗏ㠃㏆ഐࡢὶ㏿࡜ᦶ᧿㏿ᗘࡢẚࢆ⾲ࡋ㸪ᗏ㠃ቃ⏺᮲s㸪b࡛Ỉ㠃࡜ᗏ㠃ࡢ್ࢆ⾲ࡍ㸬ࣞ࢖ࣀࣝࢬᛂຊࡣὶ㏿㖄௳ࢆ୚࠼ࡿ㸬୍⯡࡟ࡣᗏ㠃㏆ഐ࡛ᑐᩘศᕸ๎ࡀ௬ᐃࡉࢀ
4)┤ศᕸࢆ⪃៖ࡋࡓ୍᪉⛬ᘧࣔࢹ࡛ࣝィ⟬ࡍࡿ6)㸬
ࡿ 㸬ࡋ࠿ࡋ㸪Ἑᗋ㏆ഐࡢ ᒙෆࡢὶࢀࡣ㸪LSRࡢᏑᅾ࡟ࡼࡗ࡚஘ࡉࢀࡿࡓࡵ㸪➼ὶ≧ែࡢ௬ᐃ࡟ᇶ࡙ࡃᑐᩘศ
ᘧ(1)㸪(2)ࢆィ⟬ࡍࡿࡓࡵ࡟ࡣ㸪ὶ㏿࡜ᅽຊࡢ㖄┤ศᕸ๎ࡀᡂࡾ❧ࡓ࡞࠸࡜⪃࠼ࡽࢀࡿ㸬ࡇࡢὶࢀࡢ㠀ᖹ⾮ᛶᕸ㸪ཬࡧᗏ㠃ࡢ㖄┤᪉ྥὶ㏿࡜ࡏࢇ᩿ᛂຊࢆィ⟬ࡍࡿᚲࡣ⢒ᗘᒙ࡟ࡶཬࡪࡓࡵ㸪Ẽ㇟ศ㔝ࡢᶞᮌホ౯࡞࡝࡟⏝࠸せࡀ࠶ࡿ㸬ᚋ⪅࡟ࡘ࠸࡚ࡣ㸪ḟ❶ࡢ㠀ᖹ⾮⢒㠃᢬ᢠ๎࡟ࡽࢀ࡚࠸ࡓࢲࣈࣝ࢔࣮࣋ࣞࢪࣥࢢࡢ⪃࠼᪉ࢆᑟධࡋ㸪ࡼࡾィ⟬ࡍࡿ㸬๓⪅࡟ࡘ࠸࡚ࡣ㸪୕ḟ㛵ᩘࡢὶ㏿ศᕸ(3)ᒙෆࡸ⢒ᗘᒙෆࡢ㠀ᖹ⾮ὶࢀࢆ㐠ື᪉⛬ᘧ࠿ࡽồࡵࡿ᪉࡜ᅽຊࡢ㠀㟼Ỉᅽᡂศࡢ⥺ᙧศᕸ(4)ࢆ௬ᐃࡍࡿ㸬ἲࡀᥦ᱌ࡉࢀ࡚࠸ࡿ5)㸬ࡇࡢ᪉ἲࡶ㸪ᇶ♏᪉⛬ᘧࡢᖹᆒu'i 'ui 12K 3  12K 2  1  Gui  4K 3  3K 2 (3)໬࡟క࠺✀ࠎࡢᮍ▱ಀᩘࢆỴᐃࡍࡿᚲせࡀ࠶ࡿࡇ࡜࡜㸪ᚑ᮶ࡢ᢬ᢠ๎࡜ࡢ㛵㐃ࡀ᫂☜࡛࡞࠸ㄢ㢟ࢆ᭷ࡋ࡚࠸ࡿ㸬(4)
dp Kdpbࡲࡓ㸪୕ḟඖゎᯒἲࡣィ⟬㈇Ⲵࡀ኱ࡁࡃ㸪♟ᗋἙᕝ࡬ࡢ
ࡇࡇ࡟㸪K=(zsz)/h㸪'ui = usi Ui㸪Gui= usi ubi ࡛࠶ࡿ㸬ὶ㐺⏝ᛶࡣᅔ㞴࡛࠶ࡿ㸬㏿࡜ᅽຊࡢ㖄┤ศᕸࢆồࡵࡿࡓࡵ࡟㸪BVCἲ࡛ࡣ௨ୗࡢ
ⴭ⪅ࡽࡣ㸪ỈἐࡍࡿἙᕝᵓ㐀≀࿘ࡾࡢᒁᡤὙ᥀ゎᯒࣔ᪉⛬ᘧ⣔㸸Ỉ῝✚ศࡉࢀࡓ ᗘࡢᐃ⩏ᘧ(5)࡜ ᗘ᪉⛬ᘧࢹࣝࢆᵓ⠏ࡍࡿࡓࡵ࡟㸪㟼Ỉᅽศᕸ➼ࡢὸỈὶࡢ௬ᐃࢆ
(6)㸪Ỉ⾲㠃ࡢ㐠ື᪉⛬ᘧ(7)㸪஧㔜Ỉ῝✚ศࡋࡓ㐃⥆ᘧ⏝࠸࡞࠸‽୕ḟඖゎᯒἲ(BVCἲ)ࢆ㛤Ⓨࡋ࡚ࡁࡓ6) 㸬(8)ཬࡧỈ῝✚ศࡋࡓ㖄┤᪉ྥ㐠ື᪉⛬ᘧ(9)ࡀゎ࠿ࢀࡿ㸬
BVCἲࡣἙᗋࡢพฝ࡞࡝࡟ࡼࡿᗏ㠃㏆ഐࡢὶࢀ࡜ᅽຊศ
ᕸࢆゎᯒ࡛ࡁࡿࡓࡵ㸪LSRࡀὶࢀ࡟୚࠼ࡿᙳ㡪ࢆホ౯࡛
wz
wz
wWh
(5)
 ws s  wb b
ubi u si  H ij 3: j h ࡁࡿ࡜⪃࠼ࡽࢀࡿ㸬ࡋ࠿ࡋ㸪ᚑ᮶ࡢ୕ḟඖゎᯒ࡜ྠᵝ࡟㸪
wxi
wxi
wxi
ቨ㠃㏆ഐ࡛ᖹ⾮≧ែࢆ௬ᐃࡋࡓBVCἲ࡛ࡣ㸪Ỉ⾪㒊ࡢ஧ḟὶᙉᗘࡀᙅ࠸➼㸪♟ᗋἙᕝ࡟࠾ࡅࡿᒁᡤὶゎᯒࡢ⢭ᗘ
whDZijw: i h(6)
RVi  PZ i  wVbZbi
࡟ㄢ㢟ࡀ࠶ࡿࡇ࡜ࡀ᫂ࡽ࠿࡜࡞ࡾ7),8)㸪ᗏ㠃ቃ⏺᮲௳ࡢྲྀ
wt
wx j
8)
ࡾᢅ࠸ࡢၥ㢟ࡀ⪃࠼ࡽࢀ࡚࠸ࡓ 㸬

wu si
wu
wz
(7)
 u sj si  g * s  Psi
㸱㸬ᗏ㠃ὶ㏿ゎᯒἲwtᮏゎᯒἲ࡛ࡣ㸪ᅗ࡟♧ࡍ ᒙࡼࡾࡶୖࡢ㡿ᇦࡢὶ㏿࡜ᅽຊࡢศᕸࢆゎࡃࡓࡵ࡟BVCἲ6)ࢆ㐺⏝ࡍࡿ㸬ࡓࡔࡋ㸪㠀ᖹ⾮⢒㠃᢬ᢠ๎࡟ᛂ⏝ࡍࡿࡓࡵ㸪ᗏ㠃zb࡟࠾ࡅࡿ㖄┤ὶ㏿ࢆ⪃៖ࡍࡿ㸬 BVCἲ࡛ࡣ㸪㐃⥆ᘧ࡜ࣞ࢖ࣀࣝࢬᖹ
ᆒࡉࢀࡓ㐠ື᪉⛬ᘧࢆᗏ㠃ࡢⷧ࠸ ᒙ࠿ࡽỈ㠃ࡲ࡛✚ศࡋࡓᘧ(1)㸪(2)ࡀゎ࠿ࢀࡿ㸬

Wh

wx j

wxi

§ wz
wz · w 2h¨ m  U j m ¸ 
h k 2 'u j  k1Gu j (8)
¨ wt
¸ wx
w
x
j ¹
j
©

^

dpb

U

wz whW zj
whW whWU j W bj b 
wx j
wt
wx j
wx j

`

(9)

ࡇࡇ࡟㸪:i㸸Ỉ῝ᖹᆒ ᗘ㸪W㸸Ỉ῝ᖹᆒ㖄┤᪉ྥὶ㏿㸪- 218 -

z➼ὶ䛾ὶ㏿ศᕸ

Zb wVb
u
bi
zb
uvi
wVt
zt uri㠀ᖹ⾮䛾ὶ㏿ศᕸ

Wbi
ᖹ⾮≧ែ䛾
ὶ㏿ศᕸ

Wti

Di
xi

୺ィ⟬
㡿ᇦ
ᒙ

W bi
U

§ wui ·
¨ vt¸
© wz ¹ b

vtb  Abubi  uvi
h

(13)

W ti
U

§ wui ·
¨ vt¸
© wz ¹t

vtt  Atuvi  u rih(14)ಀᩘAb㸪At ࡣᖹ⾮≧ែ࡟࠾࠸࡚㸪Wb=Wt h/(h+Gzb)=Uu*2㸪
vtb=vtt=Du*h࡜࡞ࡿࡼ࠺࡟㸪

Gzb

⢒ᗘᒙ
Gzt

1
Ab

D (cb  cv ) 㸪 At1h  Gzb (15)D ( cv  c r ) hᅗ 㠀ᖹ⾮⢒㠃᢬ᢠ๎࡜⾲ࡍ㸬ࡇࡇ࡟㸪cb=ub/u*㸪cv=uv/u*㸪cr=ur/u*࡛࠶ࡿ㸬cb㸪
cvࡣᑐᩘศᕸ๎ࢆ⏝࠸࡚㸪
RVi =usiZsV ubiZbV㸪ZsV㸪ZbV㸸usi㸪ubiࡢᅇ㌿㸪PZi㸸 ᗘ⏕⏘㡯㸪DZij㸸⛣ὶ㸪ᅇ㌿㸪஘ὶᣑᩓ࡟ࡼࡿ ᗘࣇࣛࢵ1 § Gz b  Gz 0 ·(16)¸¸  Arcbln¨ࢡࢫ㸪g*㸸Ỉ㠃ࡢ㖄┤᪉ྥᅽຊ໙㓄㸪Psi㸸Ỉ⾲㠃ὶ㏿⏕
N ¨© k s¹⏘㡯㸦Ỉ㠃ࡢࡈࡃⷧ࠸ᒙࡢୗ㠃࡟స⏝ࡍࡿࡏࢇ᩿ᛂຊ㸧㸪࡛࠶ࡿ㸬
1 § Gz b / 2  Gz 0 ·(17)¸¸  Ar
cv
ln¨¨


N

㸲㸬㠀ᖹ⾮⢒㠃᢬ᢠ๎

©

ks

¹

࡜⾲ࡍ㸬crࡣ㸪ᘧ(12)ࡢᢠຊ㡯ࢆ⏝࠸࡚㸪ᚑ᮶ࡢ⢒㠃᢬ᢠ๎࡛ࡣ㸪ᅗ࡟♧ࡍᗏ㠃ᐃ⩏఩⨨zbࡼ
Uu r 2 § h  Gz r  Gz b ·
2
(18)
cr
ࡾୗࡢᒙ ᒙ㸪⢒ᗘᒙ ࡟࠾࠸࡚㸪ᗏ㠃ὶ㏿࡟ᑐࡍࡿᖹ
¨
¸
D ©
h
¹
⾮≧ែࡢὶ㏿ศᕸࡀ௬ᐃࡉࢀ㸪ᗏ㠃ࡢቃ⏺᮲௳ࡀ୚࠼ࡽ
࡜ᐃ⩏࡛ࡁࡿ㸬ࡓࡔࡋ㸪D 2=DiDi ࡛࠶ࡿ㸬ᢠຊ㡯ࡣ㸪⢒ࢀࡿ㸬ࡋ࠿ࡋ㸪LSR࡟ࡼࡗ࡚ᗏ㠃㏆ഐࡢὶࢀሙࡣ஘ࡉࢀ
ᗘ⢏Ꮚ࡟స⏝ࡍࡿὶయຊࢆᢠຊಀᩘCDࢆ⏝࠸࡚⾲ࡋ㸪ࡿ㸬౛࠼ࡤ㸪ᅗ࡟♧ࡍࡼ࠺࡟ᗏ㠃ࡼࡾୖࡢὶ㏿ศᕸࡀ
SC D
LSR࡟ࡼࡗ࡚ኚᙧࡋ࡚ࡶᗏ㠃ὶ㏿ࡀྠࡌ࡛࠶ࢀࡤ㸪ᚑ᮶(19)
Di
Uu r u riࡢ⢒㠃᢬ᢠ๎࡛ࡣྠࡌὶ㏿ศᕸࡀ௬ᐃࡉࢀࡿၥ㢟ࡀ࠶ࡿ㸬8࡜⾲⌧ࡍࡿ㸬ᮏ◊✲࡛ࡣ㸪CD=0.4ࢆ⏝࠸ࡿ㸬࡞࠾㸪CDࡢࡑࡇ࡛㸪ᮏ◊✲࡛ࡣ㸪 ᒙ࡜⢒ᗘᒙ࡟࠾࠸࡚㐃⥆ᘧ(10)್ࡀゎᯒ⤖ᯝ࡟୚࠼ࡿᙳ㡪ࡣᑠࡉ࠸㸬࡜㐠ື᪉⛬ᘧ(11)㸪(12)ࢆゎࡃ㸬
wGzb uvi
wOGzt u ri
(10)
, wVt
wVb wVt 㸳㸬㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡢ㐺⏝⠊ᅖࡢ⪃ᐹ

wxi

wxi



wuvi
wu
 uvk viwt
wxk

w (dpb  Ugz s ) W biW
 ti (11)
Uwxi
UGzb UGzb

wuriwu urk ri
wt
wxk



D
w (dpb  Ugzs ) W ti i (12)
Uwxi
UGzr UGzr

ࡇࡇ࡟㸪k =1,2,3(zV)㸪Gzb,Gzt㸸 ᒙ㸪⢒ᗘᒙࡢཌࡉ㸪O㸸
⢒ᗘᒙࡢ✵㝽⋡㸪Wbi,Wti㸸ᗏ㠃㸪⢒ᗘᒙୖ㠃࡟స⏝ࡍࡿࡏࢇ᩿ᛂຊ㸪Di㸸⢒ᗘᒙෆࡢ⢏Ꮚ࡟స⏝ࡍࡿᢠຊ㡯࡛࠶ࡿ㸬ࡲࡓ㸪ୗ௜ࡢv㸪rࡣ ᒙ࡜⢒ᗘᒙࡢ್ࢆ⾲ࡍ㸬ࡇࢀࡽࡢ᪉⛬ᘧࡣ ᒙ㸪⢒ᗘᒙࡀ༑ศⷧ࠸࡜௬ᐃࡍࢀࡤᐜ
᫆࡟ᑟ࠿ࢀࡿࡀ㸪ࡏࢇ᩿ᛂຊ㡯ࡸᢠຊ㡯ࢆ࡝ࡢࡼ࠺࡟ホ౯ࡍࡿ࠿ࡀ㔜せ࡜࡞ࡿ㸬ᮏ◊✲࡛ࡣ㸪ᖹ⾮≧ែ࡟࠾࠸࡚
ᚑ᮶ࡢ᢬ᢠ๎ࢆ‶ࡓࡍࡼ࠺࡟ᑟ࠿ࢀࡿ㸬ᚑ᮶ࡢ᢬ᢠ๎࡛
ࡣ㸪ᘧ(10)࡜ᘧ(11)㸪(12)ࡢ⛣ὶ㡯࡜㠀㟼Ỉᅽᡂศࡢ㡯ࡀ↓どࡉࢀࡿ㸬
ᗏ㠃㸪⢒ᗘ⾲㠃࡟స⏝ࡍࡿࡏࢇ᩿ᛂຊࡣ ື⢓ᛶಀᩘࢆ⏝࠸࡚⾲ࡍ㸬㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡢᚲせᛶ࡜㐺⏝⠊ᅖࢆ᫂ࡽ࠿࡟ࡍࡿ
ࡓࡵ㸪ෆ⏣࣭⚟ᒸ6)ࡢ㐠ື᪉⛬ᘧ(2)࡜ᗏ㠃ὶ㏿᪉⛬ᘧ(5)࡟ࡘ࠸࡚ࡢ᳨ウ࡜ྠᵝ࡟㸪ὸỈࣃ࣓࣮ࣛࢱHs=h0/L0 (h0㸸௦⾲Ỉ῝㸪L0㸸ᑐ㇟࡜ࡍࡿὶࢀሙࡢࢫࢣ࣮ࣝ)ࢆ⏝࠸㸪࡝ࡢࡼ࠺࡞᮲௳࡟࠾࠸࡚㸪㠀ᖹ⾮⢒㠃᢬ᢠ๎ࢆᵓᡂࡍࡿ
㐠ື᪉⛬ᘧࡢ࡝ࡢ㡯ࡀᨭ㓄ⓗ࡜࡞ࡿࡢ࠿ࢆᇶ♏᪉⛬ᘧࡢྛ㡯ࡢ࣮࢜ࢲ࣮ࢆẚ㍑ࡍࡿ㸬ᅗ࡟㸪኱ࡁ࡞⢒ᗘ(ks/h=0.1)࡜ᑠࡉ࡞⢒ᗘ(ks/h=0.01)
ࡢሙྜࡢ㸪ᑐ㇟࡜ࡍࡿὶࢀሙࡢࢫࢣ࣮ࣝࡢኚ໬࡟ᑐࡍࡿ
ᒙࡢ㐠ື᪉⛬ᘧࡢྛ㡯ࡢ኱ࡁࡉࡢኚ໬ࢆ♧ࡍ㸬ὸỈࣃ࣓࣮ࣛࢱHsࡀᑠࡉ࠸࡯࡝኱ࡁ࡞ࢫࢣ࣮ࣝࡢ⌧㇟㸪኱ࡁ࠸࡯࡝ᑠࡉ࡞㸪ࡘࡲࡾᒁᡤⓗ࡞⌧㇟ࢆ⾲ࡍ㸬⦪㍈ࡣ㸪ᖹ⾮≧ែࢆ௬ᐃࡋࡓᚑ᮶ࡢ᢬ᢠ๎ࢆᵓᡂࡍࡿ㸪㔜ຊ㡯࡜ࡏࢇ᩿ᛂຊ㡯ࡢ኱ࡁࡉࢆ1࡜ࡋࡓሙྜࡢྛ㡯ࡢ኱ࡁࡉࢆ⾲ࡋ࡚࠸ࡿ㸬ẚ㍑ࡢࡓࡵ㸪ᅗ D ࡟ࡣ㸪Ỉ῝ᖹᆒὶ㏿ሙ࡜ᗏ㠃ὶ㏿ሙ࡟ᑐࡍࡿᖹ㠃஧ḟඖゎᯒ࡜୕ḟඖゎᯒࡢ㐺⏝⠊ᅖ6)ࢆ♧ࡍ㸬Hs<0.01࡛ࡣ㸪࠸ࡎࢀࡢ⢒ᗘࡢ኱ࡁࡉ࡟࠾
࠸࡚ࡶࡍ࡭࡚ࡢ㠀ᖹ⾮㡯ࡀᑠࡉ࠸ࡇ࡜ࡀศ࠿ࡿ㸬ࡇࢀࡣ㸪- 219 -

3.0

3.0

⛣ὶ㡯

⛣ὶ㡯
2.5

Ỉ῝ኚ໬㡯

ྛ㡯ࡢ኱ࡁࡉ

ྛ㡯ࡢ኱ࡁࡉ

2.5

㠀㟼Ỉᅽศᕸ㡯

2.0

2D(ᗏ㠃ὶ㏿)
1.5

3D

2D(Ỉ῝ᖹᆒὶ㏿)

1.0

Ỉ῝ኚ໬㡯
㠀㟼Ỉᅽศᕸ㡯

2.0

1.5

2D(ᗏ㠃ὶ㏿)
2D(Ỉ῝ᖹᆒὶ㏿)㔜ຊ㡯䛸ᗏ㠃䛫䜣᩿ຊ㡯㔜ຊ㡯䛸ᗏ㠃䛫䜣᩿ຊ㡯

0.5

0.5

0.0
1.0EͲ03

1.0EͲ02

1.0EͲ01

1.0E+00

1.0E+01Hs

0.0
1.0EͲ03

1.0EͲ02(a) ks/h=0.01

1.0E+00

1.0E+01

1.0E+00

1.0E+01

Hs

3.0
⛣ὶ㡯

⛣ὶ㡯

2.5

2.5

Ỉ῝ኚ໬㡯

Ỉ῝ኚ໬㡯
2.0

㠀㟼Ỉᅽศᕸ㡯ྛ㡯ࡢ኱ࡁࡉ

ྛ㡯ࡢ኱ࡁࡉ

1.0EͲ01(a) /ks/h=0.01

3.0

1.5

1.0
㔜ຊ㡯䛸ᗏ㠃䛫䜣᩿ຊ㡯

0.5

2.0

㠀㟼Ỉᅽศᕸ㡯1.5

1.0㔜ຊ㡯䛸ᗏ㠃䛫䜣᩿ຊ㡯

0.5

0.0
1.0EͲ03

1.0EͲ02

1.0EͲ01

1.0E+00

1.0E+01Hs

0.0
1.0EͲ03(b) ks/h=0.1
ᅗ
(m)

3D

1.0

10.2 10.4 10.6 10.8 11 11.2 11.4(m)

2 (m/s)㠀ᖹ⾮ᛶࡀ኱ࡁࡃ࡞ࡿࡢ࡟ᑐࡋ㸪⢒ᗘᒙෆ࡛ࡣ⢒ᗘࡀ኱ࡁ࠸࡯࡝㠀ᖹ⾮ᛶࡀ኱ࡁࡃ࡞ࡿ㸬኱ࡁ࠸⢒ᗘ(ks/h=0.1)ࡢ
ሙྜ࡛ࡣ㸪 0.1<Hs࠿ࡽ⢒ᗘᒙෆࡢ㐠ື᪉⛬ᘧࡢࡍ࡭࡚ࡢ㡯ࡀ↓ど࡛ࡁ࡞ࡃ࡞ࡿ㸬ࡇࡢࡇ࡜ࡣ㸪౛࠼ࡤᗏ㠃㏆ഐࡢゎᯒ࣓ࢵࢩࣗࡀ༑ศᑠࡉ࠸୕ḟඖゎᯒ࡟࠾࠸࡚ࡶ㸪ᑡ࡞ࡃ࡜ࡶ⢒ᗘ⾲㠃ࡼࡾୗ࡛ᖹ⾮≧ែࢆ௬ᐃࡍࡿᚑ᮶ࡢ⢒㠃᢬ᢠ๎ࡣ㐺⏝࡛ࡁ࡞ࡃ࡞ࡿࡇ࡜ࢆ♧ࡋ࡚࠸ࡿ㸬⢒ᗘᒙෆࡢὶࢀࡢ㠀ᖹ⾮ᛶࡀ↓ど࡛ࡁ࡞ࡃ࡞ࡿὶࢀࡢࢫࢣ࣮ࣝࡣ
୕ḟඖゎᯒࡀᚲせ࡜࡞ࡿࢫࢣ࣮ࣝࡼࡾࡶ኱ࡁࡃ㸪ᴫࡡ‽୕ḟඖゎᯒࡀᚲせ࡜࡞ࡿࢫࢣ࣮ࣝ⛬ᗘ࡛࠶ࡿࡓࡵ㸪⢒ᗘࡀ኱ࡁ࠸Ἑ㐨࡟࠾࠸࡚ὶࢀࡢ୕ḟඖᛶࢆ⪃៖ࡍࡿᚲせࡀ࠶ࡿሙྜ࡟ࡣ㸪㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡀᚲせ࡜࡞ࡿ࡜ゝ࠼ࡿ㸬


ㆤᓊ

ᕧ▼

4
6
8
10

g
110

105

f

e
100

Hs

ᅗ ⢒ᗘᒙࡢ㐠ື᪉⛬ᘧࡢྛ㡯ࡢኚ໬02

1.0EͲ01(b) ks/h=0.1ᒙࡢ㐠ື᪉⛬ᘧࡢྛ㡯ࡢኚ໬

-2

1.0EͲ02

d c

b
95

a
90
(m)

ᅗ ᖖ㢪ᑎᕝ⌧ᆅᐇ㦂࡟࠾ࡅࡿỈ⾪㒊௜㏆ࡢἙᗋᙧ≧㸪ὶ㏿ ᐃ᩿㠃࡜Ỉ㠃࡜ᗏ㠃㏆ഐὶ㏿ࡢ ᐃ⤖ᯝ7)኱ࡁ࡞ࢫࢣ࣮ࣝࡢゎᯒ࡟࠾࠸࡚ࡣᚑ᮶ࡢ⢒㠃᢬ᢠ๎࡛༑
ศ࡛࠶ࡿࡇ࡜ࢆ♧ࡋ࡚࠸ࡿ㸬ࡋ࠿ࡋ㸪ὸỈࣃ࣓࣮ࣛࢱHs
ࡀᑠࡉࡃ࡞ࡿ࡜㸪㠀ᖹ⾮㡯ࡀ኱ࡁࡃ࡞ࡾ㸪ᚑ᮶ࡢ⢒㠃᢬ᢠ๎ࡀ㐺⏝࡛ࡁ࡞࠸ࡇ࡜ࡀศ࠿ࡿ㸬≉࡟㸪ᑠࡉ࠸⢒ᗘ(ks/h=0.01)ࡢሙྜ࡛ࡣ㸪 0.01<Hs࠿ࡽ ᒙෆࡢ㠀ᖹ⾮ᛶࡀ↓ど࡛ࡁ࡞ࡃ࡞ࡿ㸬ࡇࢀࡣ ᒙࡢཌࡉ࡟ᑐࡋ࡚⢒ᗘࡀᑠࡉ࠸࡯࡝㠀ᖹ⾮ᛶࡀᙉࡃ࡞ࡿࡇ࡜ࢆ♧ࡋ࡚࠸ࡿ㸬୕ḟඖゎᯒ࡟࠾࠸࡚ࡣ㸪Ἑᗋ࡟᥋ࡍࡿ࣓ࢵࢩࣗࡢཌࡉࡀ ᒙࡢཌࡉ࡟ᑐᛂࡋ࡚࠾ࡾ㸪ᑠࡉ࡞⢒ᗘࡢゎᯒ࡯࡝Ἑᗋ㏆ഐ࡛⣽࠿࡞࣓ࢵࢩࣗࡀᚲせ࡜࡞ࡿ㸬BVCἲ࡛ࡣ ᒙࡢཌࡉGzbࡣỈ῝hࡢ⣙1/20ಸ࡛ᐃ⩏ࡉࢀ࡚࠸ࡿ㸬ゎᯒἲࡢ㐺⏝⠊ᅖ6)࡜ẚ㍑ࡍࡿ࡜㸪 ᒙෆࡢὶࢀࡢ㠀ᖹ⾮ᛶࡣᴫࡡ2D
࡜3Dࡢ㛫㸪༶ࡕ‽୕ḟඖゎᯒἲࡀᚲせ࡜࡞ࡿὶࢀࡢࢫ
ࢣ࣮ࣝ࠿ࡽ㔜せ࡜࡞ࡿ㸬ᅗ࡟⢒ᗘᒙࡢ㐠ື᪉⛬ᘧࡢྛ㡯ࡢ኱ࡁࡉࡢኚ໬ࢆ♧ࡍ㸬ᅗࡢ ᒙࡢሙྜ࡜ẚ㍑ࡍࡿ࡜㸪㠀㟼Ỉᅽ㡯ࡣᴫࡡྠࡌ࡛࠶ࡾ㸪ࡑࡢ࡯࠿ࡢ㡯ࡣ ᒙࡢሙྜࡼࡾࡶᑠࡉࡃ
࡞ࡗ࡚࠸ࡿ㸬ࡋ࠿ࡋ㸪 ᒙෆ࡛ࡣ⢒ᗘࡀᑠࡉࡃ࡞ࡿ࡯࡝

㸴㸬♟ᗋἙᕝ࡟࠾ࡅࡿᒁᡤὶ࡬ࡢ㐺⏝ᮏ◊✲࡛㛤Ⓨࡋࡓ㠀ᖹ⾮⢒㠃᢬ᢠ๎ࢆᑟධࡋࡓᗏ㠃ὶ㏿ゎᯒἲ(BVC-NWL)ࢆ2011ᖺᖖ㢪ᑎᕝ⌧ᆅᐇ㦂9)࡟㐺⏝ࡋ㸪㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡢ♟ᗋἙᕝ࡟࠾ࡅࡿᒁᡤὶሙࡢゎᯒ࡬ࡢጇᙜᛶࢆ᳨ドࡍࡿ㸬2011ᖺᖖ㢪ᑎᕝ⌧ᆅᐇ㦂࡛ࡣ㸪ᖖ㢪ᑎᕝ◁ᕞୖ࡟Ỉ㊰ᖜ20m㸪పỈ㊰ᖜ5m㸪పỈ㊰Ỉ῝
0.7mࡢ」᩿㠃⺬⾜Ỉ㊰ࡀసᡂࡉࢀࡓ㸬ᅗ࡟♧ࡍࡼ࠺࡟పỈ㊰⺬⾜㡬㒊ࡢỈ⾪㒊࡟ࢥࣥࢡ࣮ࣜࢺ〇ࡢᆶ┤ㆤᓊࡀタ⨨ࡉࢀ㸪ࡑࡢ┤ୗ࡟ࡣᕧ▼┒ᅵ◁ᕞࡀタࡅࡽࢀࡓ㸬ᐇࢫࢣ࣮ࣝࡢ⃭ࡋ࠸ὶࢀࡢᒁᡤ୕ḟඖὶ㏿ሙࢆィ ࡍࡿࡓࡵ㸪ὶ㏿ィ ࡟ࡣADCPࡢಶูࡢὶ㏿ࣅ࣮࣒ࢹ࣮ࢱࢆ⏝࠸ࡓ୕ḟඖὶࢀࡢィ ἲ7)ࡀ⏝࠸ࡽࢀࡓ㸬ᐇ㦂㸪ィ ἲࡢヲ⣽࡟ࡘ࠸࡚ࡣᩥ⊩7),9)ࢆཧ↷࡟ࡉࢀࡓ࠸㸬ゎᯒࡢቃ⏺᮲௳࡟ࡣ㸪ὶ㔞ࡼࡾࡶ⢭ᗘࡀ㧗࠸࡜⪃࠼ࡽࢀࡿỈ఩ࢆୖୗὶ࡟୚࠼ࡓ7)㸬ᗏ㠃⢒ᗘࡣἙᗋ⾲ᒙ࡟Ꮡᅾࡍࡿ♟ᚄࡢ⠊ᅖෆ࡛㸪ᐇ ࡢỈ఩⦪᩿ศᕸࡸὶ㔞㸪ὶ㏿ศᕸࢆㄝ᫂ࡍࡿࡼ࠺࡟ks =0.08mࢆ⏝࠸㸪ᕧ▼タ⨨ಶᡤ- 220 -Electromagnetic

(m)

11.6
11.4
11.2
11
10.8
10.6
10.4

ㆤᓊ

revetmentᐇ ⤖ᯝ7)

Section C

5
11.6
11.4
11.2
11
10.8
10.6
10.4

4

3

2

1

11.6
11.4
11.2
11
10.8
10.6
10.4

2D
5

0 (m)
11.6
11.4
11.2
11
10.8
10.6
10.4

4

3

2

1

0 (m)

4

3

2

1

0 (m)

4

3

2

1

0 (m)

4

3

2

1

0 (m)

4

3

2

1

0 (m)

3

2

1

0 (m)

(m)

(m)

㟁☢ὶ㏿ィ
current meterᖹᆒỈ఩Mean water level஧ḟὶ࣋ࢡࢺࣝ㸸 2.0(m/s)

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2(m)

୺ὶ㏿ (m/s)㸸

BVC-EWL
5

4

3

2

1

BVC-NWL
50 (m)D ᩿㠃 crevetment

11.6
11.4
11.2
11
10.8
10.6
10.4

ᐇ ⤖ᯝ7)

Section E

5
11.6
11.4
11.2
11
10.8
10.6
10.4

4

3

2

1

2D
5

0 (m)
11.6
11.4
11.2
11
10.8
10.6
10.4

(m)

(m)

Electromagnetic
current meter

Mean water level

(m)

(m)

11.6
11.4
11.2
11
10.8
10.6
10.4

BVC-EWL
Section E
5

4

3

2

1

BVC-NWL
Section E5

0 (m)ᐇ ⤖ᯝ7)
5

4

3

2

1

0 (m)

11.4
11.2
11
10.8
10.6
10.4
10.2

11.4
11.2
11
10.8
10.6
10.4
10.2

2D
5

(m)

(m)

11.4
11.2
11
10.8
10.6
10.4
10.2

(m)

E ᩿㠃 eBVC-EWL
Section F
5

4

3

2

1

0 (m)

11.4
11.2
11
10.8
10.6
10.4
10.2

BVC-NWL
Section F
54

F ᩿㠃 fᅗ Ỉ⾪㒊ࡢ୕ḟඖὶ㏿ሙࡢᐇ ⤖ᯝ࡜ྛ✀ゎᯒἲ(2D㸪BVC-EWL㸪BVC-NWL)࡟ࡼࡿゎᯒ⤖ᯝࡢẚ㍑

ࡣks = 0.2mࢆ୚࠼࡚࠸ࡿ㸬ゎᯒᆅᙧࡣ⦪᩿ᖜ0.5m㸪ᶓ᩿ᖜ࡟஧ḟὶࡀᙧᡂࡉࢀ࡚࠸ࡿ㸬2Dゎᯒ⤖ᯝ࡛ࡣ஧ḟὶࡢ
0.25mࡢࢹ࢝ࣝࢺᗙᶆ࣓ࢵࢩࣗ࡟5.0mẖࡢTS 㔞⤖ᯝ࡜ᙳ㡪ࡀ⪃៖ࡉࢀࡎ㸪ᐇ ⤖ᯝ࡜ẚ㍑ࡍࡿ࡜୺ὶ㏿ࡀෆᓊ
ADCPἙᗋᙧ≧ ᐃ⤖ᯝࢆ⏝࠸࡚సᡂࡋࡓ7)㸬ࡲࡓ㸪ᖹ
ࡼࡾ࡟Ⓨ⏕ࡋ㸪పỈ㊰ᗏ㠃௜㏆ࡢὶ㏿ࡢపୗࡀ⾲⌧࡛ࡁ㠃஧ḟඖゎᯒἲ(2D)࡜ᖹ⾮⢒㠃᢬ᢠ๎ࢆ⏝࠸ࡓᗏ㠃ὶ㏿࡚࠸࡞࠸㸬BVC-EWLゎᯒ⤖ᯝ࡛ࡣ㸪ᐇ ⤖ᯝ࡜ྠᵝ࡟ゎᯒἲ(BVC-EWL)࡟ࡼࡿྠ୍᮲௳ࡢゎᯒࢆ⾜࠸㸪BVC஧ḟὶࡀィ⟬ࡉࢀ㸪஧ḟὶ࡟ࡼࡿୖ㏙ࡢ୺ὶ㏿ศᕸࡢኚ
NWL࡜ẚ㍑ࡋ㸪㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡢ᭷ຠᛶࢆ᳨ウࡍࡿ㸬 ᙧࢆㄝ᫂ࡋ࡚࠸ࡿ㸬ࡋ࠿ࡋ㸪஧ḟὶࡢ኱ࡁࡉࡀᐇ ⤖ᯝᅗ࡟ᅗ࡟♧ࡍ᩿㠃c,e,f࡟࠾ࡅࡿ୕ḟඖὶ㏿ሙࡢẚ࡜ẚ࡭ࡿ࡜࠿࡞ࡾᑠࡉࡃ㸪୺ὶ㏿ศᕸࡢኚᙧࡶ༑ศ࡛࡞
㍑ࢆ♧ࡍ㸬᩿㠃c,eࡢᐇ ⤖ᯝ࡛ࡣADCP࡛ィ ࡛ࡁ࡞࠸
࠸㸬BVC-NWLゎᯒ⤖ᯝ࡛ࡣ㸪ᐇ ⤖ᯝ࡜ẚ࡭ࡿ࡜ㆤᓊ
9)Ỉ㠃௜㏆࡟࠾࠸࡚㟁☢ὶ㏿ィࡢ୺ὶ㏿ࡢ ᐃ⤖ᯝ ࡶྜ
ቨ㏆ഐ࡛஧ḟὶࡀᙅ࠸ࡶࡢࡢ㸪BVC-EWL࡜ẚ㍑ࡍࡿ࡜
ࢃࡏ࡚♧ࡋ࡚࠸ࡿ㸬᩿㠃cࡢᐇ ⤖ᯝ࡛ࡣ㸪పỈ㊰඲య஧ḟὶᙉᗘཬࡧ஧ḟὶ࡟ࡼࡿ୺ὶ㏿ศᕸࡢኚᙧࡢゎᯒ⤖- 221 -

ᯝࡀᨵၿࡉࢀ࡚࠸ࡿࡇ࡜ࡀศ࠿ࡿ㸬ᕧ▼௜ࡁ┒ᅵ◁ᕞࡢ┤ୖὶ࡛࠶ࡿ᩿㠃e࡛ࡣ㸪ᐇ ⤖ᯝࢆぢࡿ࡜㸪᩿㠃c࡜ẚ
㍑ࡋ࡚ᙉ࠸஧ḟὶࡀᙧᡂࡉࢀ㸪ᗏ㠃㏆ഐ࡛ࡣෆᓊ࡟ྥ࠿
࠺ᙉ࠸ὶࢀࡀⓎ⏕ࡋ࡚࠸ࡿ㸬BVC-EWLゎᯒ⤖ᯝ࡛ࡣ2D
ゎᯒ࡛ࡣ⪃៖࡛ࡁ࡞࠸ࡇࡢᐇ ⤖ᯝࡢ≉ᚩࢆㄝ᫂ࡋ࡚࠸
ࡿࡀ㸪᩿㠃c࡜ྠᵝ࡟஧ḟὶࡀᙅࡃィ⟬ࡉࢀ࡚࠸ࡿ㸬ࡇࢀ࡟ᑐࡋ㸪BVC-NWLゎᯒ⤖ᯝࢆぢࡿ࡜㸪ᐇ ⤖ᯝ࡜ྠ⛬ᗘࡢᗏ㠃ὶ㏿ࡀィ⟬ࡉࢀ㸪పỈ㊰୰ኸ௜㏆ࡢ஧ḟὶࢭࣝ୰ኸࡢୗ㝆ὶࡀ෌⌧࡛ࡁ࡚࠸ࡿ㸬᩿㠃f࡛ࡣ㸪ᐇ ⤖ᯝ࠿ࡽᕧ▼⩌ࢆ஌ࡾ㉺࠼ࡓ᪩࠸ὶࢀࡢ㐠ື㔞ࡀᕧ▼⩌ࡢ᢬ᢠ࡟ࡼࡾෆᓊ᪉ྥ࡟ୗᚋὶࢆక࠸࡞ࡀࡽ㍺㏦ࡉࢀ㸪ᕧ▼⩌๓㠃࡛ࡣᗏ㠃㏆ഐ࡟㏿࠸୺ὶ㏿࡜ෆᓊ᪉ྥ࡬ࡢᙉ࠸
ᶓ᩿᪉ྥὶ㏿ࡀⓎ⏕ࡋ࡚࠸ࡿࡇ࡜ࡀศ࠿ࡿ㸬ࡇࡢὶࢀ࡟
ࡼࡗ࡚㸪ᅗ࡟♧ࡍࡼ࠺࡞ᕧ▼⩌๓㠃ࡢᒁᡤὙ᥀ࡀᙧᡂ
ࡉࢀࡿ㸬BVC-EWLゎᯒ⤖ᯝࡣ㸪2Dゎᯒ⤖ᯝ࡜ẚ㍑ࡋ࡚
ᨵၿࡉࢀ࡚࠸ࡿࡶࡢࡢ㸪ᕧ▼⩌ୖࡢෆᓊ᪉ྥ࡟ྥ࠿࠺ὶ
ࢀࡢୗྥࡁᡂศࡀᙅࡃ㸪ᕧ▼⩌๓㠃ࡢᗏ㠃௜㏆࡬ࡢ㐠ື㔞㍺㏦ࡀ༑ศ࡛࡞࠸㸬ࡲࡓ㸪పỈ㊰୰ኸ௜㏆ࡢ஧ḟὶࡀᙅࡃ㸪Ỉ㠃௜㏆࡛ࡶෆᓊ᪉ྥࡢὶࢀ࡜࡞ࡗ࡚࠸ࡿ㸬ࡇࢀࡣᖹ⾮⢒㠃᢬ᢠ๎࡛ࡣᕧ▼⩌ࡢࡼ࠺࡞኱ࡁ࡞⢒ᗘࡢᙳ㡪ࢆ༑ศ࡟ホ౯ࡍࡿࡇ࡜ࡀ࡛ࡁ࡞࠸ࡓࡵ࡛࠶ࡿ㸬BVCNWL࡛ࡣୖ㏙ࡢBVC-EWLࡢゎᯒ⤖ᯝࡢㄢ㢟ࢆᨵၿࡋ㸪ᕧ▼㏆ഐࡢ⃭ࡋ࠸ᐇ㦂ࡢὶࢀࢆ⾲⌧࡛ࡁ࡚࠸ࡿ㸬᪤ ◊✲࡛ࡣ㸪ᮏᐇ㦂࡟ᑐࡍࡿᒁᡤὶࡢゎᯒ⢭ᗘࡢၥ㢟࡜ࡋ࡚㸪᥋㏆ὶ㏿ࡢ෌⌧⢭ᗘࡢၥ㢟㸪༶ࡕゎᯒ᮲௳ࡢタᐃࡀཎᅉࡢ୍ࡘ࡟⪃࠼ࡽࢀ࡚࠸ࡓ7)ࡀ㸪ᮏㄽᩥ࡛♧ࡋࡓࡼ࠺࡟⢒㠃ࡢຊᏛᶵᵓࢆ㐠ື᪉⛬ᘧ࡜㐃⥆ᘧ࡟ᇶ࡙࠸
࡚⪃៖ࡍࡿࡇ࡜࡛㸪♟ᗋἙᕝࡢᒁᡤὶࡀ࠿࡞ࡾࡢ⛬ᗘㄝ࡛᫂ࡁࡿࡇ࡜ࡀ᫂ࡽ࠿࡜࡞ࡗࡓ㸬㸵㸬⤖ㄽ
ᮏ◊✲࡟ࡼࡾᚓࡽࢀࡓ୺せ࡞⤖ㄽࢆ௨ୗ࡟♧ࡍ㸬1) ᵝࠎ࡞኱ࡁࡉࡢ⢒ᗘࡀᏑᅾࡍࡿ♟ᗋἙᕝࢆゎᯒࡍࡿࡓࡵ࡟㸪⢒ᗘࡢከ㔜ࢫࢣ࣮ࣝᛶࢆᑟධࡋࡓ᪂ࡋ࠸ゎᯒ᪉ἲ(BVC-NWL)ࢆ㛤Ⓨࡋࡓ㸬ゎᯒ࡛┤᥋ホ౯࡛ࡁࡿἙ
ᗋࡢพฝ࡛࠶ࡿ኱ࡁ࡞⢒ᗘ(LSR)࡟ᑐࡋ࡚BVCἲࢆᛂ⏝ࡋ㸪ᑠࡉ࡞⢒ᗘ(SSR)ࡢ᢬ᢠ࡟ᑐࡋ࡚ࡣ㸪ᗏ㠃㏆ഐࡢⷧ࠸ ᒙ࡜⢒ᗘ⾲㠃ࡢ⢒ᗘᒙࡢ㐠ື᪉⛬ᘧ࡟ᇶ࡙ࡃ
㠀ᖹ⾮⢒㠃᢬ᢠ๎(NWL)ࢆᑟ࠸ࡓ㸬2) 㠀ᖹ⾮⢒㠃᢬ᢠ๎ࢆᵓᡂࡍࡿ㐠ື᪉⛬ᘧࡢྛ㡯ࡢ኱ࡁࡉࡢẚ㍑࠿ࡽ㸪ᗏ㠃㏆ഐࡢⷧ࠸ ᒙཬࡧ⢒ᗘᒙ࡟࠾ࡅࡿὶࢀࡢ㠀ᖹ⾮ᛶࡸᅽຊࡢ㠀㟼Ỉᅽᡂศࡣ㸪㠀㟼Ỉᅽࡢ‽୕ḟඖゎᯒ࡜ྠᵝ࡟Ỉ῝ࡼࡾࡸࡸ኱ࡁ࡞ὶࢀࡢ
ࢫࢣ࣮ࣝ࠿ࡽ኱ࡁࡃ࡞ࡿࡇ࡜ࢆ᫂ࡽ࠿࡟ࡋࡓ㸬⢒ᗘࡀ኱ࡁࡃ࡞ࡿ࡯࡝ ᒙෆࡢ㠀ᖹ⾮ᛶࡣᙅࡃ࡞ࡿࡀ㸪⢒ᗘᒙෆࡢ㠀ᖹ⾮ᛶࡣᙉࡃ࡞ࡿࡓࡵ㸪⢒ᗘࡢ኱ࡁ࠸♟ᗋἙᕝࡢᒁᡤὶゎᯒ࡟࠾࠸࡚ࡣ㠀ᖹ⾮⢒㠃᢬ᢠ๎ࡣ㔜せ࡞ᙺ๭ࢆᣢࡘࡇ࡜ࢆ᫂ࡽ࠿࡟ࡋࡓ㸬3) ᖖ㢪ᑎᕝ⌧ᆅᐇ㦂࡟࠾ࡅࡿỈ⾪㒊ࡢ୕ḟඖὶࢀࡢ

BVC-NWLゎᯒ࡛ࡣ㸪BVC-EWLゎᯒ࡛ࡣ༑ศ࡟⾲⌧࡛ࡁ࡞࠿ࡗࡓỈ⾪㒊ࡢᙉ࠸஧ḟὶࡸࡑࢀ࡟క࠺୺ὶ㏿ࡢኚ໬࡟ࡘ࠸࡚ᨵၿࡉࢀࡓ㸬≉࡟㸪ᕧ▼⩌㏆ഐࡢᗏ㠃௜㏆ࡢὶࢀࢆㄝ࡛᫂ࡁࡿࡇ࡜࠿ࡽ㸪♟ᗋἙᕝࡢᒁᡤὶ
ࡸᒁᡤὙ᥀ࡢゎᯒ࡟࠾࠸࡚ࡣ㸪‽୕ḟඖゎᯒἲ࡟ࡼࡿ
ὶࢀࡢ୕ḟඖᛶࢆホ౯ࡍࡿ࡜࡜ࡶ࡟㠀ᖹ⾮⢒㠃᢬ᢠ๎ࢆᑟධࡍࡿࡇ࡜ࡀ㔜せ࡛࠶ࡿࡇ࡜ࢆ᫂ࡽ࠿࡟ࡋࡓ㸬
ᮏㄽᩥ࡛㏙࡭ࡓࡼ࠺࡟BVC-NWLἲࡣὙ᥀ࢆᘬࡁ㉳ࡇࡍᗏ㠃㏆ഐࡢὶࢀࢆㄝ࡛᫂ࡁࡿࡇ࡜࠿ࡽ㸪ᒁᡤὙ᥀ゎᯒࡢ⢭ᗘྥୖࡀᮇᚅ࡛ࡁࡿ㸬ᮏ㠀ᖹ⾮⢒㠃᢬ᢠ๎ࢆἙᗋኚື
ゎᯒ࡟ᑟධࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪ᚑ᮶ࡢゎᯒἲࡀᣢࡘ஧ࡘ
ࡢㄢ㢟ࢆᨵၿࡍࡿࡇ࡜ࡀ࡛ࡁࡿ࡜⪃࠼ࡽࢀࡿ㸬ᚑ᮶ࡢᒁᡤὙ᥀ゎᯒἲࡢከࡃࡣὙ᥀῝ࢆᑠࡉࡃぢ✚ࡶࡿㄢ㢟ࢆ᭷ࡋ࡚࠸ࡓ㸬➨୍ࡢ࣓ࣜࢵࢺࡣἙᗋᴟ㏆ഐࡢ㠀ᖹ⾮ὶࢀ࡟ࡼࡿ௜ຍⓗ࡞ࡏࢇ᩿ᛂຊࢆ⪃៖࡛ࡁࡿࡇ࡜࡛࠶ࡿ㸬➨஧ࡢ࣓ࣜࢵࢺࡣ㸪Ἑᗋ㏆ഐὶ㏿࡛࡞ࡃ㸪Ἑᗋᮦᩱ࡟స⏝ࡍ
ࡿὶ㏿ሙࢆᤲὶຊィ⟬࡟⏝࠸ࡿࡇ࡜ࡀ࡛ࡁࡿⅬ࡛࠶ࡿ㸬༶ࡕ㸪ᗏ㠃ࡏࢇ᩿ຊࢆ௓ࡍࡿࡇ࡜࡞ࡃ㸪Ἑᗋᮦᩱ⢏Ꮚ࡟స⏝ࡍࡿὶయຊࢆ┤᥋ィ⟬࡛ࡁࡿࡓࡵ㸪ᤲὶຊࡢ⢭ᗘྥୖࡔࡅ࡛࡞ࡃ㸪ὶࢀࡢゎᯒἲࡀ┤᥋ⓗ࡟ࣜࣥࢡࡋࡓ᪂ࡓ
࡞ᅵ◁ࡢ㠀ᖹ⾮㐠ືゎᯒἲࡢ㛤Ⓨ࡟᭷฼࡛࠶ࡿ㸬௒ᚋࡣ
BVC-NWLἲࢆᛂ⏝ࡋࡓὶࢀ࡜ᅵ◁㐠ືࡢ୍యゎᯒἲࢆ
㛤Ⓨࡋ㸪ᒁᡤὙ᥀ゎᯒ࡟ᛂ⏝ࡍࡿணᐃ࡛࠶ࡿ㸬ཧ⪃ᩥ⊩1) Clifford, N. J., Robert, A. and Richards, K. S: Estimation offlow resistance in gravel-bedded rivers: A physical explanation of the multiplier of roughness length, Earth Surface Processes and Landforms 17(2): 111-126, 1992.
2) ྜྷᕝ⚽ኵⴭ㸪ὶ◁ࡢỈ⌮Ꮫ㸪୸ၿ㸪1985㸬3) ෆ⏣㱟ᙪ㸪⚟ᒸᤩ஧㸪⚟ᓥ⌶஧㸪⏣୰ṇᩄ㸸኱ᆺ⢒ᗘ⩌ୖࡢὸ࠸ὶࢀࡢᖹ㠃஧ḟඖゎᯒ࡜ࡑࡢᛂ⏝㸪ᅵᮌᏛ఍ㄽᩥ㞟㸪No.691/II-57㸪pp.93-103㸪2001㸬
4) Nicholas, A.P.: Computational fluid dynamics modelling ofboundary roughness in gravel-bed rivers: an investigation of
the effects of random variability in bed elevation, Earth Surf.
Process. Landforms 26: 345-362, 2001.
5) Nikora, V., McEwan, I., McLean, S., Coleman, S., Pokrajac,
D. and Walters, R.: Double-Averaging Concept for RoughBed Open-Channel and Overland Flows: Theoretical
Background, J. Hydraul. Eng 133: 873-883, 2007.
6) ෆ⏣㱟ᙪ㸪⚟ᒸᤩ஧㸸ὸỈὶࡢ௬ᐃࢆ⏝࠸࡞࠸Ỉ῝✚ศࣔࢹࣝ࡟ࡼࡿᗏ㠃ὶ㏿ࡢゎᯒἲ㸪ᅵᮌᏛ఍ㄽᩥ㞟
B1(ỈᕤᏛ)㸪Vol.68㸪No.4㸪I_1225-I_1230㸪2012㸬
7) ㍿▼኱㸪ෆ⏣㱟ᙪ㸪⚟ᒸᤩ஧㸸ㆤᓊỈ⾪㒊࡟࠾ࡅࡿ୕
ḟඖὶࢀ࡜Ἑᗋᙧ≧ࡢほ ἲ࡜ゎᯒἲࡢ㛤Ⓨ㸪ᅵᮌᏛ఍ㄽᩥ㞟 B1(ỈᕤᏛ)㸪Vol.69㸪No.4㸪I_1171-I_1176㸪
2013.
8) ㍿▼኱㸪ᖹሯ┿⌮Ꮚ㸪ෆ⏣㱟ᙪ㸪⚟ᒸᤩ஧㸸ㆤᓊỈ⾪㒊࡟࠾ࡅࡿ୕ḟඖὶࢀࡢゎᯒἲࡢᨵⰋ࡜ㄢ㢟,Ἑᕝᢏ⾡ㄽᩥ㞟, ➨ 19
ᕳ, pp.99-104, 2013㸬9) ᑠụ⏣┿௓㸪▼஭㝧㸪ᒾ஭ஂ㸪▼ᕝಇஅ㸪⚟ᒸᤩ஧㸸Ỉ⾪㒊ᑐ⟇ࢆ᪋ᕤࡋࡓ◁ᕞ࡟ࡼࡿ⮬↛ᛶࡢ㧗࠸Ἑᓊ㜵ㆤᕤࡢ๰ฟ㸪Ἑᕝᢏ⾡ㄽᩥ㞟㸪➨ 18 ᕳ㸪pp.233-238㸪2012㸬㸦ཷ௜㸧- 222 -